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GIFT  TO  THE   LIBRARY 


Civil  Engineering  Department 

UNIVERSITY    OF    CALIFORNIA 

BY 

PROFESSOR  FRANK  SOULE 
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! 


UNIVERSITY  OF  CALITOrr 

DEPARTMENT  OF  CIVIC  ENGINEER! NO 
Owfi.;.  :<.£•,  UA  IFOR, 


Digitized  by  the  Internet  Archive 

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THE   ELEMENTS 


OF 


LOGARITHMS 


WITH  AN  EXPLANATION  OF  THE 


THREE  AND  FOUE  PLACE  TABLES 


OF   LOGARITHMIC    AND    TRIGONOMETRIC 
FUNCTIONS 


BY 


JAMES   MILLS   PEIRCE 

UNIVERSITY    PROFESSOR    OP    MATHEMATICS    IN    HARVARD    UNIVERSITY 


BOSTON 

PUBLISHED    BY  GINN    BROTHERS 

1873 


>4 


Engineering 
Library 


:ering 
library 


Entered  according  to  Act  of  Congress,  in  the  year  1873,  by 

JAMES    MILLS    REIRCE, 

In  the  Office  of  the  Librarian  of  Congress,  at  Washington. 


camhridp.e: 
press  of  john  wilson  and  son. 


PREFACE. 


r  I  ^HIS  little  book  is  meant  chiefly  for  beginners ; 
but  I  have  also  had  in  view  the  wants  of 
more  advanced  students  who  are  seeking  to  refresh 
and  deepen  their  knowledge.  The  sections  included 
in  brackets  are  inserted  for  their  benefit;  and  it  is 
my  purpose  to  add,  in  a  future  edition,  chapters  on 
the  natural  system,  series,  and  the  errors  incident  to 
simple  interpolation,  so  as  to  make  the  book  a  com- 
plete manual  of  all  that  can  be  considered  as  belong- 
ing to  the  elements  of  the  subject. 

Logarithms  ought  not  to  be  comprised,  as  they 
often  are,  in  the  midst  of  a  treatise  on  algebra.  For, 
in  the  first  place,  they  are  not  algebraic  functions; 
and,  besides  this,  the  student  is  unlikely  to  form  an 
adequate  comprehension  of  their  purpose  or  to  appre- 
ciate the  importance  of  acquiring  skill  in  the  use  of 
his  tables,  if  he  takes  them  up  in  the  course  of  a 
study  to  which  they  have  no  application.  If  loga- 
rithms must  needs  be  combined  with  any  other  branch 
of  mathematics,  their  true  alliance,  on  grounds  both 

789544 


IV  PREFACE. 

theoretical  and  practical,  is  not  with  algebra,  but 
with  trigonometry.  But  it  seems  to  me  desirable 
that  so  important  a  subject  should  be  studied  by 
itself,  and  studied  fully  and  thoroughly;  and  the 
aim  of  the  teacher  should  be  the  double  one  of  ex- 
pounding the  doctrine  of  logarithms  in  a  concise, 
accurate,  and  clear  form,  and  of  inculcating  and 
enforcing  good  practical  principles  in  the  use  of 
tables.  It  has  been  my  endeavor,  accordingly,  in 
writing  these  pages,  not,  of  course,  to  convert  the 
learner  into  an  accomplished  computer,  but  to  set 
him  on  the  right  road  towards  becoming  such  if  he 
pleases,  and,  at  all  events,  to  lead  him  to  form  the 
sound  habits  of  work  without  which  little  beneficial 
discipline  is  derivable  from  this  study. 

I  hope  that  teachers  who  are  preparing  pupils  for 
Harvard  College,  where  logarithms  are  now  required 
for  admission,  will  find  that  this  volume  furnishes 
them  with  a  satisfactory  text-book.  The  course  I 
would  recommend  for  such  students  comprises  all  that 
precedes  the  Appendix,  except  the  bracketed  portions. 
Those  teachers  who  have  not  time  to  go  over  so  much 
ground  will  find  it  possible  to  omit  also  §§  7,  8,  13, 
15,  17,  18,  and  43,  and  perhaps  some  other  sections. 
Those,  on  the  other  hand,  who  are  able  to  exceed 
the  course  described  above  are  advised  to  take  up 
the  table  of  Logarithms  of  Sums  and  Differences. 

Only  the  rudiments  of  the  theory  of  logarithms 


PREFACE.  V 

are  required  at  the  examination,  but  candidates  are 
expected  to  be  thoroughly  drilled  in  practical  work. 
Applicants  in  Course  II.,  however,  should  be  well 
prepared  on  all  the  unbracketed  parts  of  the  book, 
including  the  Appendix.  The  use  of  the  trigono- 
metric tables  is  not  required  for  admission  in 
Course  I. 

I  would  further  respectfully  ask  the  attention  of 
professors  of  mathematics  in  other  colleges  to  the 
desirableness  of  putting  into  the  hands  of  students 
tables  of  three  or  four  places  instead  of  the  bulky 
volumes  which  even  practical  computers  need  for 
only  the  more  delicate  kinds  of  work,  and  especially 
instead  of  the  inconvenient  and  poorly  constructed 
tables  often  found  bound  up  in  the  same  covers  with 
a  work  on  trigonometry.  This  has  been  pointed  out 
by  several  of  the  best  authorities  abroad,  and  is,  no 
doubt,  widely  acknowledged  here  as  well.  I  venture 
to  hope,  therefore,  that  the  publication  of  this  little 
treatise  will  induce  a  more  general  employment  of 
the  tables  to  which  it  is  designed  as  a  companion. 

J.  M.  Peiece. 

Cambridge.  October  1873. 


CONTENTS. 


Page 

CHAPTER  I. —  General  Principles 1 

CHAPTER  H.  — The  Denary  System 17 

CHAPTER  HI.  —  Computation  by  Logarithms     .    .  26 

Table  of  Proportional  Parts 37 

Three-Place  Table  of  Logarithms  of  Numbers  ...  38 

Four-Place  Table  of  Logarithms  of  Numbers     ...  39 

Arithmetical  Complement.     Reciprocal 43 

Multiplication  by  Logarithms 44 

Division  by  Logarithms 45 

The  Rule  of  Three  by  Logarithms 46 

Involution  by  Logarithms 46 

Evolution  by  Logarithms 47 

Compound  Operations 48 

Exponential  Equations 48 

Logarithms  of  Sums  and  Differences 49 

APPENDIX.  —  Explanation  of  the   Trigonometric 

Tables 55 

Three-Place  Table  of  Trigonometric  Functions  ...  56 
Four-Place  Table  of  Logarithmic  Trigonometric  Func- 
tions       63 

Inverse  Trigonometric  Functions 69 

Examples  in  Trigonometry  adapted  to  the  Four-Place 

Tables 76 


THE 


ELEMENTS   OF  LOGARITHMS. 


CHAPTER  I. 

GENERAL     PRINCIPLES. 

§  1*  Logarithms  are  numbers  that  are  substituted  in 
computation  for  other  numbers,  to  which  they  bear  such 
a  relation  that  the  operations  to  be  performed  on  the 
latter  are  represented  by  simpler  operations  performed  on 
the  former.  The  method  of  logarithms  was  first  pro- 
pounded in  1614,  in  a  book  entitled  Mirifici  Logarith- 
morum  Canonis  Description  by  John  Napier  (latinized 
Neperus),  Baron  of  Merchiston  in  Scotland,  who  was 
born  about  1550,  and  died  in  1618,  four  years  after  the 
publication  of  his  memorable  invention.  This  method 
has  contributed  very  powerfully  to  the  modern  advance 
of  science,  and  especially  of  astronomy,  by  facilitating 
the  laborious  calculations  without  which  that  advance 
could  not  have  been  made.  It  is  also  constantly  employed 
in  surveying,  navigation,  and  other  branches  of  practical 
mathematics ;  and  it  may  be  used  with  much  advantage 
in  all  multiplications  and  divisions  of  numbers  of  three 
or  more  figures. 

Besides  their  usefulness  in  computation,  logarithms 
fill  an  important  place  in  the  higher  theoretical  mathe- 
matics ;  but,  in  this  book,  we  shall  consider  them  almost 
wholly  in  their  practical  aspiect. 


1  THE    ELEMENTS    OF    LOGARITHMS. 

The  word  logarithm,  which  is  due  to  Napier  himself, 
is  formed  from  Xo'/o^,  ratio,  and  df/idfiog,  number,  and 
means  a  number  that  indicates  a  ratio.  It  refers  to  the 
proposition,  which  will  be  proved  in  its  proper  place,  and 
which  was  made  by  Napier  his  fundamental  theorem, 
that  the  difference  of  two  logarithms  determines  the 
ratio  of  the  numbers  for  which  they  stand,  so  that  an 
arithmetical  series  of  logarithms  corresponds  to  a  geo- 
metric series  of  numbers. 

§  So  The  base  of  a  system  of  logarithms  is  a  fixed 
number  to  which  all  numbers  are  referred  in  that  system. 
It  may  have  any  positive  value  except  unity. 

The  logarithm  of  a  number  in  any  system  is  the 
exponent  of  the  power  to  which  the  base  of  the  system 
must  be  raised  to  produce  that  number. 

The  antilogarithm  of  a  number  is  that  number  of 
which  the  given  number  is  the  logarithm;  in  other 
words,  it  is  that  power  of  the  base  of  which  the  given 
number  is  the  exponent. 

Thus,  in  a  system  of  logarithms  of  which  8  is  the 
base, 

log        8  =  1,  antilog  1  =        8, 

log      G4  =  2,  antilog  2  =      G4, 

log    512  =  3,  antilog  3  =    512, 

log  4096  =  4,  &c,  antilog  4  =  4096,  &c. 

In  general,  if  x  and  u  have  any  such  values  as  to 
satisfy  the  equation 


we  have,  in  the  system  of  which  a  is  the  base, 

u  =  log  x,  x  =  antilog  u ; 

and  either  of  the  two  latter  equations  may  be  regarded  as 
equivalent  to  the  former. 


GENERAL   PRINCIPLES.  3 

Napier  at  first  called  logarithms  artificial  numbers, 
and  antilogarithms  natural  numbers  ;  and  the  latter  desig- 
nation is  still  often  used.  The  word  antilogarithm  is 
modern ;  and,  though  a  convenient  term,  is  used  by  a  few 
writers  only. 

§3.  The  following  formulas  are  proved  in  algebra, 
and  the  proofs,  which  are  readily  derived  from  the  prin- 
ciple that  the  exponent  of  a  quantity  denotes  the 
number  of  times  the  quantity  is  taken  as  a  factor  in 
the  term  into  which  it  enters,  should  here  be  given  by 
the  student:  — 

au  av  aw =  au+v+w+ ,  (1) 

aw 

—  =  au~v,  (2) 

aP  v  } 

(au  )v  =  a™,  (3) 

„  u 

Vau  =  a».  (4) 

§  4-.  The  following  principles  are  less  familiar :  — 
If,  in  (2),  u  =  v, 
au 
av  ' 

/.        rf  =  l;  (5) 

that  is,  the  zero  power  of  any  quantity  is  unity.  This 
result  conforms  to  the  principle  cited  in  §  3.  For  to 
introduce  any  quantity  into  a  term  zero  times  as  a  factor 
is  equivalent  to  introducing  unity  as  a  factor,  since  either 
of  these  operations  would  leave  the  term  unchanged. 
That  is,  a°b  =  5,  since  b  is  multiplied  by  a  zero  times, 
and  also  lb=b;  so  that  a°b  =  lb,  or  a0  =  1. 
If,  in  (2),  u  =  0,  then,  by  (5)  and  (2), 

au         1 

—  =  — ,  au~v  =  a~v : 

/.        a-v  =  — ;  (6) 


4  THE    ELEMENTS    OF   LOGAEITIIMS. 

that  is,  a  quantity  with  a  negative  exponent  denotes 
the  reciprocal*  of  the  same  quantity  with  the  corre- 
sponding positive  exponent.  Thus,  8~2  =  j%.  This  re- 
sult may  also  be  brought  into  conformity  with  the  prin- 
ciple of  §  3.  For  to  divide  by  av,  or  to  multiply  by  — 
is  to  take  away  a  as  a  factor  v  times;  that  is,  to  take  it  v 
times  less  than  before ;  and  this  is  precisely  what  must  be 
meant  by  taking  it  ( —  v)  times  more  than  before.  Hence 
—  is  the  same  thins?  as  a~v  . 

If,  in  (4),  u  is  not  divisible  by  v,  the  fractional  ex- 
ponent may  still  be  used  to  denote  the  operation  of 
successively  raising  to  the  power  denoted  by  the  numer- 
ator and  extracting  the  root  denoted  by  the  denominator. 

Thus,  8^  =  V82  =  4,  8*  =  J.  This  result,  again,  is  con- 
formable to  our  general  principle.  For  Va"  is  that 
quantity  which,  taken  v  times  as  a  factor,  gives  a  taken 
u  times  as  a  factor.     We  may  then  regard  it  as  a  taken 

7  times  as  a  factor,  and  hence  write  Wau  =  av . 

Exponents  may  then  be  either  positive,  negative,  or 
null,  and  may  be  either  integral  or  fractional.  They  may 
even  be  incommensurable^  for  the  value  of  a  power  hav- 
ing an  incommensurable  exponent  can  be  found,  within 
any  required  limits  of  error,  by  taking  a  sufficiently  ap- 
proximate fractional  value  of  the  exponent. 

*  The  reciprocal  of  a  quantity  is  the  quotient  obtained  by  dividing 
unity  by  that  quantity;  e.g.,  the  reciprocal  of  x  is  — . 

f  Two  quantities  are  said  to  be  incommensurable  to  each  other,  when 
they  have  no  common  measure.  A  number  is  called  incomnu  araradfa,  when 
it  is  incommensurable  to  unity,  and  therefore  cannot  be  exactly  expressed 
by  an  integer,  or  by  the  ratio  of  two  integers;  i.e.,  by  a  fraction.  Circulating 
decimals  must  not  be  confounded  with  incommensurable  numbers.  Thus, 
4/2  is  incommensurable,  but  O.G  is  not  incommensurable.  The  ratio  of  two 
quantities  incommensurable  to  each  other  is  an  incommensurable  number. 
An  incommensurable  number  can  be  expressed  fractionally  within,  any  re- 
quired limit  of  accuracy.  Thus,  L4142  expresses  y2  within  ±0.00005.  In- 
commensurable numbers  are  also  called  surds. 


GENERAL   PRINCIPLES.  D 

It  is  easily  proved  that  formulas  (1-4)  hold  for  any 
values  of  the  exponents.  For  we  have  shown  that  the 
general  principle  from  which  those  formulas  are  derived 
may  be  extended  to  all  exponents.  For  example,  we 
have 


VaVa  Va 

§  5.  Since  any  number  may  be  an  exponent,  any  num- 
ber may  be  a  logarithm. 
Thus,  if  8  is  the  base, 

8-2  =  &,  8*  =  4,        8"*  =  J. 

.«.— 2  =  ]og&      f  =  log4,     -§=logi. 

§  6.  Since  a0  =  1,  therefore, 

in  any  system,     log  1  =  0,      antilog  0  =  1.       (7) 
Since  a1  =  a,  therefore, 

in  any  system,    log  base  =  1,   antilog  1  =  base.     (8) 
By  (6),  if  a  is  the  base, 

or  the  logarithm  of  the  reciprocal  of  any  number  in  any 
system  is  equal  to  the  negative  of  the  logarithm  of  the 
number  in  the  same  system. 

If  a  >  1,  the  result  obtained  by  taking  it  as  a  factor 
an  infinite  number  of  times  is  itself  infinite ;  so  that 

a+«  =oo,«-°  =^=4-  =0.* 

*  If  x  increases  indefinitely,  -  decreases  indefinitely ;  and  if  x  decreases 
indefinitely,  —  increases  indefinitely.    Hence 

1=0,    L=oo. 
co  0 

1* 


Ill 


THE    ELEMENTS    OP   LOGARITHMS. 

A  log  oo  =  +  oo ,        antilog  (+  oo )  =  oo ,  \  ~q. 

log    0  =  —  oo ,         antilog  ( —  oo  )  =  0.     ) 

If  a  <  1,  a—1  >  1,  and  hence 
(a-i  )  +•  =a-»  =oo, a+-  =-_  =4  =  0. 

A  log  oo  =  —  oo ,         antilog  (+  oo )  =  0,     \  ^. 

log    0  =  -j-  oo ,         antilog  ( —  oo  )  =  oo  .  ) 

§  7.  If  the  base  is  greater  than  unity  (as  is  the  case 
„  the  common  systems),  its  positive  integral  j)owers  are 
all  greater  than  unity,  and  their  reciprocals,  the  negative 
integral  powers,  are,  consequently,  all  less  than  unity. 
For  example,  53  >  1,  5~3  <  1,  &c. 

771 

Any  fractional  power  of  the  base,  a11 ,  is,  in  like  manner, 
greater  or  less  than  unity,  according  as  its  nth  power,  am, 
is  greater  or  less  than  unity  (provided  n  is  positive)  ;  and 
am  js  greater  or  less  than  unity,  according  as  m  is  positive 
or  negative ;  that  is,  according  as  the  original  exponent 
—  is  positive  or  negative.  Hence,  all  the  positive  powers 
of  the  base  (both  integral  and  fractional)  are  greater  than 
unity,  and  all  the  negative  powers  are  less  than  unity ;  so 
that,  if  the  base  is  greater  than  unity,  the  logarithms  of 
all  numbers  greater  than  unity  are  positive,  and  the  loga- 
rithms of  all  numbers  less  than  unity  are  negative. 

It  is  shown  in  the  same  way  that,  if  the  base  is  less 
than  unity,  the  logarithms  of  all  numbers  less  than  unity 
are  positive,  and  the  logarithms  of  all  numbers  greater 
than  unity  are  negative. 

Thus,  if  8  is  the  base,  log  4  =  +  h  log  J  =  —  § ;  but 
if  £  is  the  base,  log  4  =  —  §,  log  \  =  +  }. 

§  8.  If  x  and  (x  -\-  h)  are  two  numbers,  and  u  and 
(u  -\-  k)  their  logarithms,  then 
x  =  au ,  x  +  h  =  au  +  k  =  au  ak  by  (1),  h  =  au  (ak  —  1). 


GENERAL   PRINCIPLES.  7 

If  then  x  -f-  h  >  x ;  that  is.  if  h  is  positive ;  ak  must  be 
greater  than  1 ;  so  that,  by  §  7,  if  a  >  1,  Jc  is  positive,  and 
w  -J-  k  >  w,  but  if  a  <[  1,  &  is  negative,  and  u  -f-  k  <^  u. 

Hence,  the  greater  of  two  numbers  has  the  greater 
logarithm  in  a  system  of  which  the  base  is  greater  than 
witty,  and  the  less  logarithm  in  a  system  of  which  the 
base  is  less  than  unity. 

Thus,  if  8  is  the  base,  log  8  >  log  4 ;  but  if  \  is  the 
base,  log  8  <  log  4. 

[§  O.*  We  have  seen  that  rt»  =  l,  if  w  =  0,  and  that 
au  differs  from  1,  if  u  differs  from  0.  But  it  is  possible  to 
make  the  difference  between  au  and  1  less  than  any 
quantity  that  may  be  assigned  by  making  u  sufficiently 
small. 

For  if  m  is  any  value  of  u  (positive,  if  a  >  1 ;  negative, 
if  a  <  1),  and  if  am  =/>>  1,  then  if  we  reduce  the  value 
of  u  to  Jm,  x  =  ahm  =pi  =  V p.  Now,  by  the  principle 
a*  —  V*  =  (a  +  b)  (a  —  b), 

But  since  p  >  1,  \fp  -f-  1  >  2. 

.-.  v!p-i<J(i>-1). 

Hence,  by  halving  the  value  of  u,  the  difference  (x  —  1) 
is  reduced  to  an  amount  less  than  half  its  former  value. 
Now  we  can  successively  halve  the  value  of  u  as  many 
times  as  we  please,  and  if  (a;  —  1)  were  halved  at  the 
same  time,  we  could,  by  carrying  the  process  far  enough, 
make  that  difference  less  than  any  assigned  quantity; 
then  we  can  certainly  accomplish  this,  when  (x  —  1)  is 
reduced  more  rapidly  than  by  successive  halving. 

We  have  seen  that  if  x  ^>  1,  we  can  find  a  value  of  u 
that  will  make  (x  —  1)  as  small  as  we  please.     Then  the 

*  Sections  included  in  brackets  may  be  omitted  by  the  beginner. 


8  THE    ELEMENTS    OF   LOGARITHMS. 

reciprocal  of  x,  which  is  less  than  1,  can  be  made  to  differ 
from  1  as  little  as  we  please,  since 

1  —  1  =  *=*<>  — L 

X  X  ^ 

But  the  value  of  u  for  —  is  the  negative  of  the  value  of 
u  for  x.     See  (9). 

Hence,  by  making  the  logarithm  sufficiently  near  0,  and 
either  positive  or  negative,  we  can  bring  (he  antilogarithm 
as  near  1  as  ice  please,  making  it  at  the  same  time  either 
greater  or  less  than  1  at  our  discretion.^ 

[§  10.  If  x  and  (x  -\-h)  are  two  numbers,  and  u  and 
(u  -\-  k)  their  logarithms,  then,  as  in  §  8, 

h  —  au  («*  —  1)  =x  (a*  —  1). 

But,  by  making  k  sufficiently  small,  we  can,  by  §  9, 
make  (ah  —  1)  as  small  as  we  please,  and  therefore  small 
enough  to  render  h  less  than  any  quantity  that  may  be 
assigned.  Hence,  by  making  a  sufficiently  small  change 
in  the  value  of  a  logarithm,  we  can  make  the  correspond- 
ing change  in  the  antilogarithm  as  small  as  we  please; 
and  if  the  logarithm  increases  continuously  (that  is,  by 
insensible  gradations)  from  —  go  to  -)-  go  ,  the  antiloga- 
rithm  will  also  change  its  value  continuously  ;  increasing 
from  0  to  -f-oo,  if  the  base  is  greater  than  unity ;  and 
decreasing  from  -f-oo  to  0,  if  the  base  is  less  than  unity  ; 
and,  in  either  case,  passing  through  every  intermediate 
value. 

We  see  then  that,  in  any  system  of  logarithms,  every 
positive  number  has  one  and  only  one  real  logarithm,  and 
conversely  that  every  real  number  (positive  or  negative  ) 
has  one  and  only  one  antilogarithm,  which  is  always 
p>ositive. 

Negative  numbers  have  no  real  logarithms,  and  when 
they  occur  in  logarithmic  calculations,  the  arithmetical 


GENERAL   PRINCIPLES.  9 

value  of  the  number  and  its  sign  must  be  considered  se- 
parately.] 

[§11.  The  word  real  is  used  in  the  last  article  in  a 
technical  sense,  which  is  likely  to  be  new  to  the  student. 
Heal  quantities,  or  reals,  are  whole  numbers,  ratios  be- 
tween whole  numbers  (that  is,  fractions),  and  incommen- 
surable numbers  (which  can  be  approximately  represented 
by  fractions),  and  are  either  positive  or  negative ;  that  is 
to  say,  they  are  the  ordinary  quantities  of  arithmetic  and 
elementary  algebra.  Distinguished  from  them  are  imagi- 
nary quantities  or  imaginaries.  Of  these  algebra  affords 
examples  in  the  even  roots  of  negative  quantities  (such 
as  \/— 1  ),  which  cannot  be  real,  since  no  real  quan- 
tity produces  a  negative  result  when  raised  to  an  even 
power.  In  the  elementary  mathematics,  imaginaries  ap- 
pear merely  as  absurd  expressions,  to  which  little  atten- 
tion is  paid ;  but  in  the  higher  study  of  the  relations  of 
quantity,  they  assume  great  importance.  In  the  higher 
theory  of  logarithms,  it  is  found  that  every  number 
(whether  positive,  negative,  or  imaginary)  has,  in  any 
system,  an  infinity  of  logarithms,  which  are,  in  general, 
all  imaginary  ;  but  that  if  the  number  is  real  and  positive, 
one  and  only  one  of  its  logarithms  is  real,  and  this  is  the 
logarithm  ordinarily  considered. 

Imaginary  logarithms  cannot  be  made  use  of  in  prac- 
tical computation,  and  we  shall  not  have  occasion  to 
consider  them  further.  We  only  refer  to  them  here  in 
order  to  explain  why  we  have  so  carefully  guarded  the 
language  of  the  foregoing  article.] 

[§  12.  We  have  said  that  a  real  logarithm  has  but 
one  antilogarithm,  which  is  always  positive.  Other  values 
of  the  antilogarithm  might  also  be  admitted.  For  exam- 
ple, since  ah  =  s/a  has  two  values,  we  might  consider 


10  THE   ELEMENTS    OF    LOGARITHMS. 

antilog  0.5  as  having  two  values,  one  positive  and  the 
other  negative.  But  mathematicians  have  found  it  best 
to  admit  only  one  value  of  the  antilogarithm,  even  in  that 
higher  theory  of  logarithms  to  which  allusion  has  just 
been  made ;  and  in  the  exponential  equation 


x  is  regarded  as  having  but  one  value  for  each  value  of  u, 
and  as  being  real  and  positive  whenever  u  is  real,  assum- 
ing a  to  be  real  and  positive.] 

§  13*  If  several  logarithms  form  an  arithmetical  pro- 
gression, their  antilogarithms  form  a  geometric  progres- 
sion. 

For  if  the  logarithms  are 

u>  u  -f-  k,  u  -f-  2£,  u  -\-  3&, 

and  if  we  denote  au  by  cc,  and  ak  by  A,  the  antilogarithms 
of  the  given  logarithms  will  be,  by  (1)  and  (3), 

x,  hx,  h2x,  hzX) 

It  will  be  observed  that  £,  the  constant  difference  in  the 
series  of  logarithms,  is  the  logarithm  of  h,  the  constant 
ratio  in  the  series  of  antilogarithms. 

Example.  In  the  system  of  which  8  is  the  base, 


antilog  4 

= 

4096, 

antilog  31 

= 

1024, 

antilog  2§ 

r=: 

256, 

antilog  2 

= 

64, 

antilog  \\ 

= 

16, 

antilog  f 

= 

4, 

antilog  0 

rz= 

1, 

antilog  (- 

i) 

= 

b 

antilog  (— 

-14) 

= 

tV, 

antilog  (— 

-2) 

= 

T&,  &C 

Here  fc= 

_f,A=i  = 

=  antilog 

(-§)• 

GENERAL   PRINCIPLES.  11 

[§  14L  The  terms  of  any  arithmetical  progression  of 
which  one  term  is  0  may  be  considered  as  logarithms 
of  the  corresponding  terms  of  any  geometric  progression 
of  which  one  term  is  1 ;  those  terms  being  regarded  as 
corresponding  which  are  similarly  situated  relatively  to 
the  terms  0  and  1  respectively,  and  the  base  of  the  system 
being  that  root  of  the  constant  ratio  of  the  latter  series  of 
which  the  exponent  is  the  constant  difference  of  the  former 
series. 

Let  7c  ±=  the  constant  difference  of  the  arithmetical  pro- 
gression, 
h  ±±  the  constant  ratio  of  the  geometric  progres- 
sion. 
Then  mk  and  hm  will  denote  any  two   corresponding 
terms  of  the  two  series,  and  if  we  take  «,  the  base  of  the 
system,  such  that 

k  i_ 

a  =  wh  =  hk ,  or  ak  =  A,  or  a"*  =  hmt 
we  have 

Jc  =  logh,         m7c  =  \og  hm; 
which  proves  the  proposition,  and  shows  that  the  constant 
difference  of  the  arithmetical  series  is  the  logarithm  of 
the  constant  ratio  of  the  geometric  series.] 

§  15.  Let  w,  u\  and  u"  be  three  successive  logarithms 
in  the  series  of  §  13,  and  x,  x',  and  x"  their  antilogarithms. 
Then 

u"  —  u'=.u'  —  u  =  &,  x'  —  x  =  (h  —  l)x, 
x"—x'=  (A  —  l)  hx  =  h  (x'  —  x). 

If  now  Jc  be  taken  very  near  0,  h  will  be  very  near  1,  since 
h  =  ak ;  and  consequently 

(x"  —  x1)  =  {x'  —  x)  approximately ; 
that  is,  if  the  successive  differences  of  three  logarithms  are 
equal  and  very  small,  the  differences  of  their  antiloga- 
rithms are  approximately  equal. 


12  TIIE    ELEMENTS    OF    LOGARITHMS. 

Hence  it  follows  that  if  the  successive  differences  of 
three  logarithms  are  very  small  and  not  equal,  the  differ- 
ences of  their  antilogarithms  are  approximately  propor- 
tional to  those  of  the  logarithms.  For  if  the  successive 
differences  of  the  logarithms  be  divided  into  equal  parts, 
those  of  the  antilogarithms  will  be  divided  into  the  same 
number  of  parts,  which  will  be  approximately  equal. 

We  do  not  here  stop  to  inquire  hoic  small  the  differ- 
ences must  be  in  order  that  the  error  resulting  from  the 
application  of  this  theorem  may  remain  within  definite 
limits.  That  is  a  question  which  requires  for  its  answer 
higher  principles  of  logarithms  than  those  explained  in 
this  chapter.  The  theorem  itself  is,  however,  necessary  in 
the  practical  use  of  logarithms. 

§  1G.  We  have  now  considered  the  general  nature  of 
logarithms.  Their  utility  in  computation  is  derived  from 
the  following  theorems  ;  — 

I.  The  logarithm  of  the  continued  product  of  several 
quantities  is  the  sum  of  their  logarithms. 

II.  The  logarithm  of  the  quotient  of  two  quantities  is 
the  excess  of  the  logarithm  of  the  dividend  over  that  of 
the  divisor. 

III.  The  logarithm  of  any  power  of  a  quantity  is  the 
product  obtained  by  multiplying  the  logarithm  of  the 
quantity  by  the  exponent  of  the  power. 

IV.  The  logarithm  of  any  root  of  a  quantity  is  the 
quotient  obtained  by  dividing  the  logarithm  of  the 
quantity  by  the  exponent  of  the  root. 

These  theorems  are  also  conveniently  stated  in  the 
following  formulas:  — 

I.  log  (xyz )  =  log*+logy+logz  + ,   (12) 

which  may  be  written,  if  Z  is  used  for  the  word  sum,  and 
II  for  the  word  product. 


GENERAL   PRINCIPLES.  13 

I.  \ogIIx  =  Z\ogx.  (13) 

X 

II.  log  —  =  log  x  —  log  y.  (14) 

III.  log  xn  =  n  log  x.  (15) 

n.  1 

IY.  log  Vx  =  —  log  x.  (16) 

The  proofs  of  these  theorems  are  easily  derived  from 
formulas  (1-4). 

I.  Let  u  =  log  x,  v  =  log  y,  w  =  log  2,  &c. 
Then,  if  a  is  the  base  of  the  system, 

x  =  auiy  =  av,  z  =  awt  &c. 

Then,  by  (1), 

xyz =zau  +  v  +  w  + , 

or  IIx=z  a2« . 

.-.  by  §  2,  log  II  x  =  Zu  z=  2  log  x. 

II.  With  the  same  notation,  by  (2), 

2/ 
.-.  by  §  2,  log  —  =  u  —  v  =  \ogx  —  logy. 

if 

III.  With  the  same  notation,  by  (3), 

xn   _  anut 

,\  by  §  2,  log  xn  =  nu  =  n  log  a; , 

where  it  should  be  observed  that  n  may  be  positive  or 
negative,  integral  or  fractional. 

IV.  With  the  same  notation,  by  (4), 

u 

n  _ 

Vx  =  an . 

n  ifi 

.\  by  §  2,  logV  x  =  —  =  —  log  x. 
n         n      ° 


14  THE    ELEMENTS    OF   LOGARITHMS. 

These  theorems  show  that  the  use  of  logarithms  en- 
ables us  to  replace  the  operations  of  multiplication,  divi- 
sion, involution,  and  evolution,  by  operations  respectively 
simpler. 

Multiplication  is  replaced  by  Addition. 
Division  is  replaced  by  Subtraction. 
Involution  is  replaced  by  Multiplication. 
Evolution  is  replaced  by  Division. 

§  17.  The  logarithms  of  proportional  numbers  are 
equidifferent. 

For  the  proportion 

m        p 
n  q 

gives,  by  (14), 

log  m  —  log  n  =  \ogp  —  log  q ;  (17) 

whence  is  derived  the  following  formula  for  the  logarithm 
of  the  fourth  term  of  a  proportion, 

log  q  =  log  p  -f-  log  n  —  log  m.  (18) 

This  proposition  is  given  by  Napier  as  the  First 
Proposition  of  the  theory  of  logarithms. 

§  18.  An  exponential  equation  is  easily  solved  by 
means  of  logarithms. 

For  let  the  equation  be 

bx  =  my 

where  x  is  the  unknown  quantity,  and  b  and  m  are  posi- 
tive.    Theu,  by  (15), 

log  m  =  log  bx  =  x  log  b. 


GENERAL   PRINCIPLES.  15 

[§  19.  If  b  be  made  the  base  of  a  system  of  logarithms, 
then,  by  §  2,  x  =  log  m  in  the  system  of  which  b  is  the 
base.  We  obtain,  then,  from  (19)  the  following  theo- 
rem:— 

If  the  logarithm  of  a  number  in  any  system  be  divided 
by  the  logarithm  of  a  second  number  in  the  same  system, 
the  quotient  is  the  logarithm  of  the  first  number  in  a  new 
system,  of  which  the  second  number  is  the  base. 

When  logarithms  belonging  to  different  systems  are 
brought  into  the  same  formula,  it  is  convenient  to  distin- 
guish them  by  writing  the  symbol  of  the  base  as  a  suffix 
to  the  symbol  "  log."     Thus, 

loga  m  =  log  m  in  the  system  of  which  a  is  the  base, 

log&  m  =  log  m b ; 

and  we  have 

'•-"Sr-]  (20) 

[§  30.  If  m  =  a,  loga  m  =  1,  by  (8),  and  (20)  becomes 

log&  a  s=  j^-p  or  loga  b.  log6  a  =  1 ;  (21) 

so  that,  when  we  change  from  one  system  of  logarithms  to 
another,  the  logarithm  of  the  old  base  in  the  new  system 
is  the  reciprocal  of  the  logarithm  of  the  new  base  in  the 
old  system.'] 

[§  21.  Combining  (20)  and  (21),  we  have 

log&  m  =  log&  a.  loga  m.]  (22) 

[§  22.  We  see,  by  (20)  and  (22),  that,  if  the  logarithms 
of  a  series  of  numbers  are  computed  in  any  system,  we 
can  convert  them  to  any  other  system,  by  dividing  all  the 
computed  logarithms  by  one  fixed  quantity,  namely  the 
logarithm  of  the  new  base  in  old  system,  or  by  multiply- 


16  THE    ELEMENTS    OF   LOGARITHMS. 

ing  all  the  computed  logarithms  by  one  fixed  quantity, 
namely  the  logarithm  of  the  old  base  in  the  new  system. 
Thus,  a  system  of  logarithms  founded  on  the  base  8 
can  be  converted  into  a  system  of  which  £  is  the  base,  by 
multiplying  the  logarithms  of  the  former  system  by  —  3, 
which  equals  log  8  in  the  latter  system  (since  (£)~3  =  8)  ; 
as  will  be  seen  by  the  following  table  : 


Nos. 

logs 

logs 

(base  8) 

(base  h) 

4096, 

4, 

-12, 

1024, 

3J, 

-10, 

256, 

2§, 

-    8, 

64, 

2, 

-    6, 

16, 

H, 

-   4, 

8, 

1, 

—   3, 

4, 

h 

-    2, 

2, 

h 

-    1, 

1. 

0, 

0, 

h 

-h 

1, 

h 

—  h 

2, 

h 

-1, 

3, 

A, 

-1J> 

4-] 

THE    DENARY    SYSTEM.  17 


CHAPTER  n. 


THE    DENARY     SYSTEM. 


§  33.  There  is  an  infinite  variety  of  possible  systems 
of  logarithms ;  but  only  three  have  been  actually  used  by 
mathematicians.  These  are  the  system  of  Napier,  the 
natural  system,  and  the  denary  system. 

The  denary,  or  decimal,  system  is  that  which  has  the 
number  10  for  its  base.  In  practical  computation,  it  is 
far  more  convenient  than  any  other,  and  is  now  used  ex- 
clusively, on  account  of  its  relation  to  the  ordinary  nu- 
meral system  of  arithmetic.  It  was  proposed  in  1617  by 
Henry  Briggs,  then  Professor  of  Geometry  in  Gresham 
College,  London,  and  afterwards  Savilian  Professor  of 
Geometry  at  Oxford.  This  mathematician  was  among 
the  first  to  recognize  the  importance  of  the  invention  of 
logarithms,  and  he  made  two  journeys  to  Scotland  for 
the  purpose  of  visiting  Napier,  in  consultation  with  whom 
he  formed  his  new  system,  —  a  system  distinguished  from 
the  original  one  not  only  in  being  founded  on  the  number 
10,  but  also  in  some  important  simplifications  of  theory. 

Denary  logarithms  are  sometimes  called  JBriggsian, 
and  sometimes  common,  or  vulgar. 

The  natural  system,  though  not  so  well  adapted  to 
use  in  computation  as  the  denary,  is  the  most  important 
of  all  systems  in  the  higher  theory  of  logarithms.  It  may 
be  regarded  as  a  modification  of  the  system  of  Napier, 
with  which  it  is  often  erroneously  confounded.  But  the 
original  Neperian  system  was  immediately  superseded  by 
2* 


18  THE    ELEMENTS    OF    LOGARITHMS. 

that  of  Briggs,  with  the  approval  of  Napier  himself,  and 
now  has  only  an  historical  interest. 

[§  24.  It  is  a  principle  of  the  Theory  of  Numbers  that 
a  composite  number  can  be  resolved  into  prime  factors  in 
only  one  way.  If  then  m  is  any  positive  integer,  10m  = 
2m  5m  can  only  be  formed  of  rn  2's  and  m  5's.     If,  again,  n 

is  any  positive  integer  not  a  divisor  of  m,  V10OT  =  10n 
cannot  be  an  integer.  For  the  prime  factors  of  such  an 
integer  must  be  divisors  of  10m  and  must  therefore  be  a 
certain  number  of  2's  and  5's,  which  repented  n  times 
would  give  m  2's  and  m  5's ;  but  this  is  impossible,  since 

n  is  not  a  divisor  of  m.  Again,  VlOm  cannot  be  a  com- 
mensurable fraction  irreducible  to  an  integer.  For  the 
?ith  power  of  such  a  fraction  would  itself  be  a  fraction 

in 

irreducible  to  an  integer.  Moreover,  10  ~~  n  can  be  neither 
an  integer  nor  a  commensurable  fraction ;  for,  if  it  were, 

m 

its  reciprocal,  10*  would  be  commensurable.  Hence,  if  u 
is  a  commensurable  irreducible  fraction  (positive  or  nega- 
tive), 10"  is  incommensurable;  and  hence,  conversely  if 
10"  is  commensurable,  u  is  incommensurable,  if  it  is  not 
an  integer.  In  the  denary  system,  then,  the  logarithms 
of  all  commensurable  numbers,  except  the  integral  powers 
of  10,  are  incommensurable,  and  the  antilogarithms  of 
all  commensurable  fractional  logarithms  are  incommen- 
surable^ 

§  25.  The  denary  logarithm  of  any  number  is  either 
an  integer,  or  else  consists  of  an  integer  (which  may  be  0) 
plus  a  fraction,  which  is  customarily  expressed  decimally 
to  a  greater  or  less  number  of  places.  The  integral  part 
of  the  logarithm  is  called  its  characteristic,  and  the  deci- 
mal part  its  mantissa.  Thus  the  characteristic  of  the 
logarithm  1.6875  .  .  is  1,  and   its   mantissa   is  .6875  . . ; 


THE    DENARY    SYSTEM.  19 

the  characteristic  of  0.3010  .  .  is  0,  and  its  mantissa  is 
.3010  . . . 

§  S6.  The  significant  figures  of  any  number  are  those 
figures  that  remain  when  all  zeros  at  the  beginning  and 
end  of  the  number  are  omitted.  Thus,  the  significant 
figures  of  the  numbers  81004000,  81004,  810.04,  and 
0.081004,  are  81004. 

It  will  be  seen  that  zero  is  a  significant  figure,  when  it 
occurs  between  other  significant  figures,  so  that  it  cannot 
be  omitted  without  changing  their  relative  numeral  posi- 
tion. 

§  27.  If  two  numbers  consist  of  the  same  series  of 
significant  figures,  their  logarithms,  in  the  denary  system, 
have  the  same  mantissa,  and  the  difference  of  their  char- 
acteristics is  equal  to  the  number  of  numeral  places  by 
which  the  units'  place  of  one  number  is  removed  from  the 
units'  place  of  the  other. 

For  it  is  a  principle  of  arithmetical  numeration  that 
the  operation  of  moving  the  units'  place  in  any  number  n 
places  is  equivalent  to  multiplying  the  number  by  10  ±w, 
the  exponent  being  positive  if  the  units'  place  is  moved  to 
the  right,  negative  if  it  is  moved  to  the  left.  But  to  mul- 
tiply a  number  by  10  ±»  is,  by  (13),  equivalent  to  adding 
the  integer  ±  n  to  its  logarithm ;  and  this  addition  does 
not  affect  the  mantissa  of  the  logarithm,  but  increases  or 
diminishes  its  characteristic  by  n,  the  number  of  places  by 
which  the  units'  place  has  been  removed. 
Thus,  if  we  know  that 

log        48.7  =  1.6875  . . , 
it  follows  that 

log      48700  =  4.6875  .  . , 

log        4.87  =  0.6875  . . , 

log  0.00487  =  —  3  +  .6875  . . 
=  —  2.3125  .  . . 


20  THE    ELEMENTS    OF   LOGAEITHMS. 

The  above  theorem  expresses  the  fundamental  property 
of  the  denary  system,  on  which  the  peculiar  utility  of  that 
system  depends. 

§  28.  In  any  system  of  logarithms,  log  1  =  0,  by  (7). 
We  have,  then,  in  the  denary  system,  by  §  27, 

log        10  =  1,  log  0.1        =  — 1 

log      100  =  2,  log  0.01      =  —  2 

log    1000  =  3,  log  0.001    =  — 3(*    (23) 

log  10000  =  4,  log  0.0001  =  —  4. 

&c,  &c., 

that  is,  the  logarithm  of  a  number  which  co?isists  of  a 

single  figure  1  and  cyphers  is  an  integer ',  and  is  equal  to 

the  number  of  places  by  which  the  units'  place  is  removed 

from  the  figure  1,  being  positive  or  negative,  according  as 

the  units'  place  is  on  the  right  or  left  of  the  figure  1,  and 

null  if  1  is  in  the  units'  place. 

The  principle  of  numeration  referred  to  in  §  27  shows 
that  the  numbers  here  spoken  of  are  the  integral  powers 
of  10  ;  that  is,  that 

10i=   io,  10-1  =  0.1, 

102=100,  10-2  =  0.01, 

and  (23)  may  also  be  obtained  directly  from  these  equa- 
tions, by  the  definition  of  a  logarithm,  §  2. 

§  29.  Any  positive  number  is  either  an  integral 
power  of  10,  or  else  it  lies  between  two  successive  in- 
tegral powers,  of  which  the  lower  contains  the  figure  1 
in  the  same  numeral  place  as  the  first  significant  figure  of 
the  given  number,  and  the  higher  contains  the  figure  1 
in  the  next  place  to  the  left.     For  example, 

10  <     48.7     <  100, 
0.001  <  0.00487  <0.01, 
&c. 


THE   DENARY    SYSTEM.  21 

Then,  by  §  8,  the  logarithm  of  a  number  which  is  not 
an  integral  power  of  10  lies  between  the  two  successive 
integers  which  are  the  logarithms  of  the  integral  powers 
determined  as  above ;  and  it  is  therefore  equal  to  the  less 
of  those  two  integers  plus  a  fractional  quantity  less  than 
1.  The  latter  quantity  is  the  mantissa  of  the  logarithm; 
the  integer  is  the  characteristic  of  the  logarithm. 

Hence,  the  characteristic  of  the  logarithm  of  any  num- 
ber, in  the  denary  system,  is  equal  to  the  number  of  places 
by  which  the  units'  place  is  removed  from  the  first  signifi- 
cant figure,  being  positive  if  the  units'  place  is  on  the 
right  of  the  first  significant  figure  (that  is,  if  the  number 
>  10),  negative  if  it  is  on  the  left  (that  is,  if  the  number 
<  1),  and  null  if  the  first  significant  figure  is  in  the 
units'  place  (that  is,  if  the  number  is  between  1  and 
10). 
Thus, 

characteristic  of  log 

characteristic  of  log 

characteristic  of  log 

characteristic  of  log 

characteristic  of  log 

characteristic  of  log 

characteristic  of  log 

characteristic  of  log  0.00487 
&c,  &c. 

§  30,  The  above  rule  for  detennining  the  characteristic 
makes  it  necessary  that  that  which  is  added  to  the  charac- 
teristic to  produce  the  logarithm,  that  is,  the  mantissa, 
shall  be  positive  in  all  cases,  even  when  the  logarithm,  as 
a  whole,  is  negative.  Thus,  log  0.00487  is  between  —  3 
and  —  2,  and  we  have  seen  (in  §  27)  that  it  is  either 

—  3  +  .  6875  . .        or        —  2  —  .  3125  . . 

But  our  rule  makes  the  characteristic  —  3,  and  therefore 


48700  = 

4, 

4870  = 

3, 

487  = 

2, 

48.7  = 

1, 

4.87  = 

o, 

0.487  =  - 

-1, 

0.0487  =  - 

-2, 

1.00487  =  - 

-3, 

22  THE    ELEMENTS    OF    LOGARITHMS. 

necessitates  the  adoption  of  the  first  of  these  forms.  In 
order  to  avoid  writing  the  sign  -f~  between  the  character- 
istic and  mantissa,  we  write  the  sign  —  over  the  charac- 
teristic, as  a  way  of  indicating  that  it  affects  that  figure 
only.    For  example,  we  write 

log  0.00487  ="3.6875... 
But  as  negative  characteristics  are  often  a  source  of  con- 
fusion  in  computation,  it  is  customary  to  add  10  to  every 
negative  characteristic,  and  to  write  —  10  after  the  loga- 
rithm.   Thus  we  have 

log  0.00487  =  7.6875  . .  —  10. 

The  — 10  is  often  omitted  in  writing,  and  left  to  be 
remembered. 

There  is  sometimes  no  advantage  in  adding  10  to  the 
negative  characteristic ;  but  the  student  is  advised  to  use 
this  method  in  all  cases,  for  the  sake  of  having  a  fixed 
rule. 

§  31.  The  characteristic  of  a  logarithm  depends,  as  we 
see,  wholly  on  the  position  of  the  units'  place  in  the 
antilogarithm,  and  not  at  all  on  the  significant  figures 
which  compose  it.  This  part  of  the  logarithm  is  then 
readily  found  by  inspection  of  the  number.  The  man- 
tissa, on  the  other  hand,  can  only  be  obtained  by  laborious 
calculations,  and  the  practical  computer  must  consult  his 
table  for  it.  But  this  part  of  the  logarithm  depends 
entirely  on  the  series  of  significant  figures  that  compose 
the  antilogarithm,  without  regard  to  the  position  of  the 
units'  place.  Hence,  if  a  table  is  so  made  as  to  give  the 
logarithm  of  any  number  between  tico  successive  integral 
powers  of  10,  as  100  and  1000,  it  will  give  the  logarithm 
of  any  number  whatever,  since  all  possible  series  of  sig- 
nificant figures  can  be  found  within  the  chosen  limits. 
This  is  a  great  advantage  which  the  denary  system  of 
logarithms  has  over  any  other. 


THE    DENARY    SYSTEM.  23 

But  we  can  suppose  an  infinity  of  numbers  between 
any  two  given  numbers,  as  100  and  1000,  and  it  is  impos- 
sible that  the  logarithms  of  all  these  numbers  should  be 
explicitly  contained  in  any  table.  They  can  only  be  given 
at  certain  intervals ;  but  if  these  intervals  are  made  suf- 
ficiently small,  the  logarithms  of  intermediate  numbers 
can  be  found  from  the  table  by  a  process  called  inter- 
polation, which  is  founded  on  the  principle  of  §  15,  and 
will  be  fully  explained  in  the  next  chapter.  The  greater 
the  degree  of  accuracy  with  which  the  logarithm  is  re- 
quired, the  smaller  should  be  the  ratio  of  the  intervals, 
in  the  table,  to  the  numbers  which  are  separated  by  those 
intervals.  Thus,  a  table  of  four-place  logarithms  may  be 
limited  to  the  logarithms  of  all  integers  from  100  to  1000, 
where  the  above-named  ratio  has  for  its  greatest  value 
T-J  o  ;  while  a  seven-place  table  should  include  the  loga- 
rithm of  every  integer  from  10,000  to  100,000,  where  the 
greatest  value  of  the  ratio  is  Tq (jo^« 

[§  32.  TVe  are  indebted  to  Briggs,  not  only  as  the 
author  of  the  denary  system,  but  also  as  the  founder, 
and,  in  great  part,  the  computer  of  the  tables  now  actually 
used.  In  1617,  he  published  the  first  instalment  of  his 
own  table,  containing  the  logarithms  of  all  integers  below 
1000  to  eight  places  of  decimals;  and  this  he  followed,  in 
1624,  by  his  Arithmetica  logarithm  tea,  containing  the 
logarithms  of  all  integers  from  1  to  20,000  and  from  90,000 
100,000  to  fourteen  places  of  decimals,  together  with  a 
learned  introduction,  in  which  the  theory  and  use  of  log- 
arithms are  fully  developed.  The  interval  from  20,000 
to  90,000  was  filled  up  by  Adrian  Ylacq,  a  Dutch  com- 
puter; but  in  his  table,  which  appeared  in  1628,  the  loga- 
rithms were  given  to  only  ten  places  of  decimals. 

"  The  total  number  of  errors  found  in  Vlacq,"  says  Mr. 
Glaisher,  "amounts  to  603,  which  probably  includes  very 
nearly  all  that  exist ;  this  cannot  be  regarded  as  a  great 


24  THE   ELEMENTS    OF   LOGARITHMS. 

number,  when  it  is  considered  that  the  table  was  the 
result  of  an  original  calculation,  and  that  more  than 
2,100,000  printed  figures  are  liable  to  error."  {Athenaeum, 
15  June  1872.  See  also  the  Monthly  Notices  of  the  Royal 
Astronomical  Society  for  May,  1872.)  All  the  modern 
published  tables  are  founded  on  that  of  Vlacq,  of  which 
an  edition,  containing  many  corrections,  was  issued  at 
Leipzig  in  1794  under  the  title  Thesaurus  Logarith- 
morum  Completus  by  George  Vega.  Concerning  Vega's 
table  Professor  De  Morgan  writes,  "  This  is,  no  doubt,  up 
to  this  time,  the  table  of  logarithms  ;  the  one  of  all  others 
to  which  ultimate  reference  should  be  made  in  questions 
of  accuracy."  (English  Cyclopaedia,  Arts  and  Sciences, 
Vol.  VII.,  article  "  Table,"  London,  18G1.) 

Callet's  seven-place  table  (Paris,  1795),  instead  of 
stopping  at  100,000,  gives  the  eight-place  logarithms  of 
the  numbers  between  100,000  and  108,000,  in  order  to 
diminish  the  errors  of  interpolation,  which  are  greatest 
in  the  early  part  of  the  table ;  and  this  addition  has  been 
since  generally  included  in  seven-place  tables.  But  the 
only  important  published  extension  of  Vlacq's  table  has 
been  made  by  Mr.  Sang  (1871),  whose  table  contains  the 
seven-place  logarithms  of  all  numbers  below  200,000. 

Briggs  and  Vlacq  also  published  original  tables  of  the 
logarithms  of  the  trigonometric  functions. 

Besides  the  tables  we  have  named,  we  ought  to  men- 
tion the  great  collection,  called  Tables  du  Cadastre,  which 
was  constructed  under  the  direction  of  Prony,  by  an 
original  computation,  under  the  auspices  of  the  French 
republican  government  of  the  last  century.  This  work, 
which  contains,  besides  other  tables,  the  logarithms  of  all 
numbers  up  to  100,000  to  nineteen  places,  and  of  the  num- 
bers between  100,000  and  200,000  to  twenty-four  places, 
exists  only  in  manuscript,  "in  seventeen  enormous  folios," 
at  the  Observatory  of  Paris.  It  was  begun  in  1792  (year 
ii.) ;  and  "  the  whole  of  the  calculations,  which  to  secure 


THE    DENAEY    SYSTEM.  25 

greater  accuracy  were  performed  in  duplicate,  and  the 
two  manuscripts  subsequently  collated  with  care,  were 
completed  in  the  short  space  of  two  years."  {English 
Cyclopcedia,  Biography  >  Vol.  IV.,  article  "  Prony.") 

For  further  information  on  the  subject  of  this  section, 
and  for  an  enumeration  and  description  of  tables  pub- 
lished up  to  1861,  the  student  is  referred  to  the  article 
"  Table  "  in  the  English  Cyclopcedia  already  cited.] 


26  THE    ELEMENTS    OF   LOGARITHMS. 


CHAPTER  III. 

COMPUTATION   BY   LOGAEITHMS. 

§  33.  This  chapter  on  the  practical  use  of  logarithms 
is  adapted  to  the  author's  Three  and  Four  Place  Tables 
of  Logarithmic  and  Trigonometric  Functions. 

§  34.  Quantities  are  divided  into  two  classes :  con- 
stants and  variables ;  and  variables  are  divided  into  two 
classes :  independent  variables  and  functions. 

A  constant  is  a  quantity  which  is  conceived  to  be 
restricted  to  one  or  more  fixed  values;  a  variable  is  a 
quantity  which  is  not  so  restricted,  and  (in  general)  may 
have  any  value. 

When  the  value  of  a  variable  is  regarded  as  being  as- 
sumed arbitrarily,  the  variable  is  called  an  independent 
variable;  when  it  is  regarded  as  being  determined  by 
the  values  of  other  variables,  the  variable  is  called  a  func- 
tion of  the  variables  by  which  it  is  determined. 

The  independent  variables  which  enter  into  any  ques- 
tion are  often  called  simply  the  variables. 

If,  for  instance,  in  the  equation  au  =  x,  a  denotes  the 
base  of  a  definite  system  of  logarithms,  while  x  denotes 
any  number,  and  u  its  logarithm,  a  is  a  constant,  and  x 
and  u  are  variables,  either  of  which  may  be  regarded  as 
an  independent  variable,  and  the  other  as  a  function  of  it. 
But  if  a  is  variable,  as  well  as  x  and  u,  any  two  of  the 
three  quantities  may  be  regarded  as  independent  variables, 
and  the  third  as  a  function  of  those  two.     Thus,  a  func- 


COMPUTATION   BY   LOGARITHMS.  Zl 

tion  may  depend  on  only  one  independent  variable  or  on 
any  number  of  independent  variables. 

A  logarithm  in  a  given  system  may  be  regarded  as  a 
function  of  its  antilogarithm,  or  an  antilogarithm  as  a 
function  of  its  logarithm.  But  we  shall  generally  con- 
sider the  relation  in  the  former  light,  and  speak  of  the 
antilogarithm  as  the  independent  variable,  and  of  its  loga- 
rithm as  a  function  of  that  variable ;  and  in  that  case  we 
shall  commonly  use  the  equation 

U    =    lOg    Xy 

in  which  we  shall  call  x  the  independent  variable  and  u  a 
function  of  x.  When  we  wish  to  regard  u  as  the  inde- 
pendent variable  and  a;  as  a  function  of  u,  we  shall  prefer 
to  write 

x  =  au ,     or    x  =  antilog  u,     or    x  =  log-1  u. 

The  last  form  of  notation  should  receive  the  special 
attention  of  the  student.  The  exponent  —  1  attached  to 
a  symbol  of  quantity  shows  that  the  quantity  is  to  be  used 
as  a  divisor,  instead  of  a  multiplier,  in  tho  term  into 
which  it  enters,  e.  g.,  on— hi  =  —  ;  or  if  n  =  mx,  x  = 
m—xn.  See  (6).  By  analogy,  then,  the  exponent  — 1  is 
attached  to  a  symbol  of  relation  to  denote  the  inversion 
of  that  relation  ;  so  that,  for  example,  if  u  =  log  x,  x  = 
log-1  u. 

§  35.  A  mathematical  table  is  an  orderly  arrange- 
ment of  the  values  of  some  function  for  certain  selected 
values  of  its  independent  variables.  Thus,  a  table  of 
logarithms  contains  the  logarithms  of  certain  selected 
numbers  (for  example,  of  all  integers  from  100  to  1000) 
arranged  in  the  order  of  magnitude  and  in  a  form  con- 
venient for  reference. 

The  independent  variables,  of  which  the  successive 
values  are  assumed  arbitrarily,  and  generally  at  equal 
intervals,  are  called  the  arguments  of  the  table.    A  table 


28  THE    ELEMENTS    OF   LOGARITHMS. 

of  logarithms  is  a  table  of  one  argument,  namely  the  anti- 
logarithm  ;  a  multiplication  table  is  a  table  of  two  argu- 
ments, namely  the  two  factors.  Most  of  the  Three  and 
Four  Place  Tables  are  tables  of  one  argument,  and  their 
argument  is  generally  distinguished  by  being  printed  in 
full-faced  type. 

§  3(5.  The  tabulated  values  of  either  the  argument  or 
the  function  in  any  table  are  those  that  are  explicitly  con- 
tained in  the  table.  The  difference  between  two  succes- 
sive tabulated  values  of  either  quantity  is  called  a  tabular 
difference.  The  letter  A  followed  by  the  letter  which  de- 
notes any  quantity  is  used  by  all  writers  to  denote  the 
difference  between  two  values  of  that  quantity.  We 
shall  here  employ  this  notation  to  designate  a  tabular 
difference.  Thus,  if  x  denotes  the  argument  of  a  table, 
and  u  the  function,  and  if  two  successive  tabulated  values 
of  x  are  xx  and  a?2,  and  the  corresponding  values  of  u  are 
ux  and  w2,  then  the  corresponding  tabular  differences  are 

Ax  =  x2  —  xv  Au  =  u2  —  uv 

The  tabular  difference  of  the  argument  generally  has 
&  fixed  value  throughout  any  given  table;  but  this  is  not 
always  the  case. 

When  a  value  of  any  quantity  is  intermediate  between 
two  successive  tabulated  values,  such  value  may  be  said 
to  lie  within  the  corresponding  tabular  difference. 

§  37,  A  table  maybe  used  in  two  ways:  directly  and. 
inversely.  The  direct  use  of  the  table  consists  in  finding 
the  value  of  the  function  for  an  assumed  value  of  the 
argument;  the  inverse  use,  in  finding  the  value  of  the 
argument  for  an  assumed  value  of  the  function.  In  either 
case,  if  the  assumed  value  is  tabulated,  the  required  value 
is  readily  found  by  inspection ;  but  if  the  assumed  value  is 
not  tabulated,  we  must  resort  to  interpolation. 


COMPUTATION  BY  LOGARITHMS.  29 

§  38.  We  shall  not  enter  on  the  complete  theoiy  of 
interpolation,  but  shall  confine  ourselves  to  simple  inter- 
polation, which  is  all  that  is  necessary  in  the  tables  we 
have  to  explain.  Simple  interpolation  rests  on  the  prin- 
ciple that  corresponding  values  of  the  argument  and  the 
function  lie  within  corresponding  tabular  differences,  and 
divide  those  differences  in  the  same  ratio.  This  principle, 
which  is  called  the  principle  of  proportional  parts,  is  ap- 
plicable, with  approximate  correctness,  to  a  table  of 
logarithms,  provided  the  tabular  differences  are  made 
sufficiently  small,  as  we  have  proved  in  §  15 ;  and  in  every 
properly  constructed  logarithmic  table,  the  tabular  differ- 
ences are  made  sufficiently  small  to  permit  the  application 
of  the  principle  up  to  the  limit  of  accuracy  belonging  to 
the  table. 

If,  in  any  mathematical  table  to  which  simple  interpo- 
lation is  applicable,  x1  and  x2  are  two  successive  tabulated 
values  of  the  argument,  and  u\  and  u%  the  corresponding 
values  of  the  function,  and  if  x  and  u  are  any  two  cor- 
responding intermediate  values  of  the  argument  and 
function,  then  the  principle  of  proportional  parts 
gives :  — 

u  —  Xl\        u2  —  u      u2  —  u\        Au 
x Xi        X2  —  x       X2  —  x\        Ax   ' 

whence  we  derive  the  following  formulas :  — 


(24) 


u  =  tn  +  ^-p-/lu,  (25) 

^2  — u  A  /oe\ 

x  =  x2 -^TAx'  <28) 


30  THE  ELEMENTS  OF  LOGAEITHMS. 

If,  then,  x  is  given,  u  may  be  found  by  either  (25)  or 
(26)  ;  and  if  u  is  given,  x  may  be  found  by  either  (27)  or 
(28). 

In  these  formulas,  the  first  term  of  the  second  member 
may  be  regarded  as  an  approximate  value  of  the  required 
quantity,  and  the  second  term  as  a  correction  to  be  applied 
to  that  approximate  value.  The  approximate  value  of 
the  required  quantity  is  that  tabulated  value  which  cor- 
responds to  either  of  the  two  tabulated  values  of  the 
assumed  quantity  between  which  the  given  value  lies. 
Whichever  of  these  two  values  be  taken,  the  same  result 
will  be  obtained  by  the  above  formulas ;  but  it  is  best  to 
form  the  habit  of  taking  that  tabulated  value  which  is 
nearest  to  the  given  value,  whether  above  it  or  below  it, 
because  that  course  is  preferable  in  some  abbreviated 
methods  of  interpolation  which  we  shall  presently  have 
to  explain. 

§  39.  If,  as  it  often  happens,  zfa  =  l,  (25-28)  become 

u  =  «i  -j-  (as  —  x\  )du>  (29) 

u  =  u2  —  (#2  —  x)/Ju,  (30) 

*=*> *T'  (32> 

§  40.  The  figures  of  the  correction  found  are  gener- 
ally to  be  carried  out  only  to  a  certain  numeral  place  deter- 
mined by  the  nature  of  the  table  in  which  the  correction 
is  applied.  If,  in  this  case,  the  figures  rejected  amount  to 
more  than  half  a  unit  in  the  place  of  the  last  figure  re- 
tained ;  that  is,  to  more  than  five  units  in  the  place  of  the 
first  figure  rejected ;  the  last  figure  retained  is  to  be  in- 
creased by  1 ;  for  the  figures  retained  will  thus  most  accu- 


c>£Fa.F.  l  l^CNT  OF  CIVIL  C»NG«N££RI 

u^ii..  '-u.1,  _     j -or- 

COMPUTATION   BY    LOGARITHMS.  31 

rately  represent  the  true  value  of  the  correction.  For 
example,  18.723  lies  between  18  and  19,  but  is  nearer  to 
the  latter  than  the  former,  since  it  exceeds  18.5.  Hence 
if  the  decimal  is  rejected,  the  number  should  be  repre- 
sented by  19,  rather  than  by  18.  On  the  other  hand, 
672.18  should  be  represented  by  672,  not  by  673,  if  the 
decimal  is  rejected. 

If  that  which  is  rejected  is  precisely  5  in  the  first 
rejected  place,  it  is  equally  correct  to  increase  the  last 
figure  retained  and  to  leave  it  unchanged.  Some  com- 
puters adopt  the  rule  for  this  case  of  increasing  the  last 
figure  retained  if  it  is  odd,  and  not  if  it  is  even,  so  as  to 
increase  it  in  half  the  cases  that  occur  in  the  long  run, 
and  to  leave  it  unchanged  in  the  other  half.  Accord- 
ing to  this  rule,  we  should  convert  3.5,  106.5,  24.5,  11.5 
into  4,  106,  24,  12.  But  sometimes  a  reason  for  adopt- 
ing some  different  proceeding  may  suggest  itself. 

The  precepts  of  this  section  should  be  applied  to  all 
cases  in  computation  in  which  the  figures  of  any  number 
are  rejected  beyond  a  certain  numeral  place. 


§  41.  We  shall  illustrate  what  has  been  said  in  the 
foregoing  sections  by  reference  to  the  following  fragment 
of  a  four-place  table  of  denary  logarithms  :  — 


x\  100     101     102     103     104     105     106     107     108     109 

X 

u     0000    0043    0086    0128    0170     0212    0253    0294    0334    0374 

u 

A«  1           43         43         42         42          42          41         41          40          40 

An 

The  argument  of  this  table  is  the  antilogarithm,  desig- 
nated by  jc,  and  it  is  tabulated  from  100  to  109,  its  tabular 
difference  being  constant  and  equal  to  1.  The  function 
is  the  mantissa  of  the  logarithm,  which  is  printed  without 
the  decimal-point.  Its  tabular  difference  varies  from  43 
to  40  (or,  more  properly,  from  .0043  to  .0040).     The  char- 


32  THE    ELEMENTS    OF    LOGARITHMS. 

acteristic  of  a  logarithm  can  readily  be  found  by  §  29 ;  and 
it  is  therefore  omitted  in  the  modern  tables. 

The  number  108.2  and  its  logarithm  may  be  said  to 
lie  within  the  last  tabular  differences  of  this  table,  accord- 
ing to  the  phraseology  of  §  36. 

§  42o  The  following  examples  of  interpolation  are 
adapted  to  the  little  table  of  logarithms  given  in  the  last 
section:  — 

Ex.  1.  Find  log  10.537.  Here,  by  §  29,  the  charac- 
teristic is  1,  and  we  have  only  to  find  the  mantissa,  which 
by  §  27,  is  the  same  as  that  of  105.37.  The  nearest  tabu- 
lated value  of  the  antilogarithm  is  here  105.  "We  have 
then 

xi  =  105,  ut  =  0212,  Ax  =  l,Au  =  41,  x  —  x1  =  0.37. 

The  correction  to  be  added  to  u\   is  then,  by  (29), 

0.37  X  41  =  15.17, 

of  which  the  unit-figure  is  to  be  added  to  the  fourth  figure 
of  u\  .  The  decimal  is,  then,  to  be  added  to  unknown 
figures  of  \i\  ,  and  it  is,  therefore,  useless,  and  should  be 
rejected.     We  have  then 

mantlogl05     =0212 
0.37  X  41  =     15 

mant  log  10537  =  0227  .-.  log  10.537  =  1.0227. 

Ex.  2.  Find  log  1017.95.  Here  we  adopt  102,  rather 
than  101,  as  the  nearest  tabulated  value ;  and  we  have,  by 
(30),  since  102  —  101.795  =  0.205, 

mant  log  102      =  0086 
0.205  X  43  =        9 
mant  log  101795  =  0077  .-.  log  1017.95  =  3.0077. 

Here  0.205  X  43  =  8.815,  which  is  represented  by  9, 
according  to  the  precept  of  §  40. 


COMPUTATION   BY   LOGARITHMS.  33 

Ex.  3.  Find  log  0.01068. 

mantlog    107  =  0294 

0.2.  X  41= 8 

mantlog  1068  =  0286  /.  log  0.01068  =  8.0286  —  10. 

Ex.  £.  Find  log-1  0.0157.  The  nearest  tabulated  man- 
tissa is  0170.    Hence  we  have,  by  (32), 

0170  =  mant  log  104 

i§=  _03 

0157  =  mant  log  1037  /.  log-1  0.0157  =  1.037. 

The  antilogarithm  should  generally  be  computed,  as 
in  this  instance,  to  only  one  place  beyond  the  tabulated 
figures. 

Ex.5.  Find  log-1  (7.0102  —  10). 

0086  =  mant  log  102 

0102  =  mant  log  1024  .-.  log-1  (7.0102  — 10)  =0.001024. 

§  43.  If  we  suppose  the  argument  of  a  table  to 
undergo  a  gradual  and  uniform  change  of  value,  the 
function  will,  at  the  same  time,  pass  through  a  gradual, 
but  not  generally  uniform,  change.  Thus,  if  x  in  the 
table  of  §  41  increases  at  a  uniform  rate  from  100  to  109, 
log  x  will  increase  continuously,  though  more  and  more 
slowly.  Now  the  derivative  of  a  function  is  another 
function  which  measures  the  rate  of  change  of  the  original 
function  as  compared  with  its  variable,  at  the  moment 
that  the  variable  passes  through  any  given  value ;  and 
this  secondary  function  can  always  be  found  by  methods 
of  the  higher  mathematics.  In  some  of  the  Three  and 
Four  Place  Tables,  the  value  of  the  derivative  is  given 
for  each  tabulated  value  of  the  function,  and  can  be  used 
as  a  multiplier,  instead  of  -—-,  in  the  formulas  of  interpo- 
lation,  through   half  the  tabular  difference  before   and 


34  THE   ELEMENTS    OF   LOGAEITHMS. 

after  the  tabulated  value  against  which  it  stands.  This 
depends  on  the  principle  already  cited  in  §  38;  for  it 
merely  assumes  that  the  function  varies  at  a  uniform 
rate,  as  compared  with  the  variable,  through  one  tabular 
difference. 

§  44.  The  results  obtained  by  logarithms  are  liable 
to  errors,  which  arise  from  the  use  of  only  a  certain  num- 
ber of  decimal-places  of  the  logarithms.  The  greater  this 
number  of  places,  the  greater  will  be  the  degree  of  ac- 
curacy of  the  results,  supposing  the  data  to  be  accurately 
known.  But  it  must  be  remembered  that,  in  any  practical 
computation,  the  data  themselves  are  subject  to  errors  of  a 
certain  magnitude,  resulting  from  the  imperfection  of  the 
means  by  which  they  are  obtained ;  and  it  is  a  mere  waste 
of  labor  to  employ  logarithms  which  will  enable  us  to 
carry  the  results  to  a  greater  number  of  places  than  the 
accuracy  of  the  data  warrants.  "  Of  the  misuse  of  tables," 
says  De  Morgan,  "  no  instance  is  more  common  than  that 
which  consists  in  taking  tables  of  too  many  places  of 
figures."  In  astronomy  and  geodesy,  the  precision  of  in- 
struments and  methods  justifies  the  use  of  seven-place 
tables.  But  in  ordinary  work,  such  as  that  of  navigation 
or  common  surveying,  the  accuracy  of  the  results  furnished 
by  a  four-place  table  is  as  great  as  we  need  or  can  expect 
to  attain ;  and  as  four-place  logarithms  take,  in  the  long 
run,  only  half  the  time  required  by  those  of  five  places, 
and  one-fourth  of  that  required  by  those  of  seven  places, 
it  is  desirable  to  employ  them  for  all  purposes  which  they 
will  serve.  A  four-place  table  is  also  well  adapted  to  be 
used  in  instruction,  where  the  leading  object  is  the  incul- 
cation of  principles.  But  the  student  who  wishes  to 
become  a  skilful  computer  must,  after  acquiring  the  com- 
mand of  such  a  table,  familiarize  himself  with  those  of  five, 
six,  and  seven  places  as  well. 

Three-place  logarithms  may  be  used  advantageously 


COMPUTATION  BY  LOGARITHMS.  35 

in  rapid  work  preliminary  to  a  detailed  computation,  and 
wherever  only  a  low  degree  of  accuracy  is  required. 

§  4o.  Before  entering  on  the  explanation  of  particu- 
lar tables,  we  would  call  the  attention  of  the  student  to 
a  few  general  principles  of  work. 

First,  the  beginner  in  the  use  of  tables  should  take 
pains  to  learn  the  best  methods  of  dealing  with  them, 
the  most  convenient  way  of  holding  them,  the  simplest 
order  to  be  followed  in  consulting  them,  and  the  proper 
employment  of  the  helps  to  interpolation  with  which 
they  are  provided ;  and  to  learn  these  things  he  must 
begin  by  making  a  careful  study  of  their  plan.  Secondly, 
he  must  form  the  habit  of  using  them  with  strict  atten- 
tion to  rule,  and  especially  of  interpolating  with  accuracy, 
when  the  best  possible  results  are  sought.  But,  thirdly, 
it  must  be  remembered  that  the  object  of  logarithms  is 
to  shorten  numerical  work.  This  object  is  only  fully 
accomplished,  when  the  computer  can  use  his  table 
rapidly,  and  interpolate,  both  directly  and  inversely,  at 
a  glance.  A  great  deal  of  this  facility  can  be  gained  in  a 
short  time  by  practice ;  and  the  learner  should  do  his  best 
to  acquire  it,  so  far  as  he  can  do  so  without  losing  his  habit 
of  correct  work.  We  may  add  that  the  computer  ought 
to  keep  his  eyes  open,  and  avoid  falling  into  mistakes 
from  which  a  little  common  sense  would  be  enough  to 
save  him. 

Among  points  which  belong  under  the  head  of  accu- 
racy, we  would  refer  again  to  the  precept  of  §  40  ;  that, 
when  figures  are  rejected  at  the  end  of  a  number,  amount- 
ing to  more  than  half  a  unit  in  the  place  of  the  last  figure 
retained,  the  last  figure  retained  should  he  increased  hy  1. 
Another  rule  in  which  the  student  requires  instruction  is 
that  zeros  at  the  end  of  a  decimal  should  not  be  omitted ; 
for  although  they  do  not  affect  the  value  of  the  number, 
they  show  that  its  value  is  known  to  a  certain  number 


36  THE   ELEMENTS    OF   LOGARITHMS. 

of  places.  For  a  similar  reason,  when  zero  is  the  char- 
acteristic of  a  logarithm,  it  is  always  written,  and  it  is  a 
good  habit  to  write  it  in  the  case  of  any  decimal  num- 
ber between  0  and  1.  Thus,  log  1.04,  if  we  are  using 
four-place  logarithms,  is  written  0.0170,  not  simply  .017, 
for  the  last  form  would  leave  the  fourth  decimal  figure 
altogether  in  doubt,  and  would  suggest  a  question  whether 
the  characteristic  had  been  thought  of.  Lastly,  the  stu- 
dent must  observe  that  it  is  as  great  an  offence  against 
the  spirit  of  accuracy  to  carry  out  a  result  beyond  the 
number  of  places  which  its  logarithm  justifies  as  it  is  to 
fall  short  of  that  number.  Thus,  we  have  found  (§  42, 
Ex.  4),  log-1  0.0157  =  1.037.  We  might  carry  out  the 
correction  of  the  tabulated  number  indefinitely,  obtaining 

if  =  0.309524.., 
log-*  0.0157  ==  1.03690476  .  . . 

But  this  would  be  to  restrict  too  closely  the  number  deter- 
mined by  a  four-place  logarithm.  In  fact,  the  antiloga- 
rithm  of  0.0157  .  .  may  lie  anywhere  between 

1.03669  .  .  =  log-1  0.0156500  .  . 
and  1.03693  .  .  =  log-1  0.0157500  .  . 

If  we  assume  that  the  value  of  the  given  logarithm  is 
0-0157000  .  .  ,  its  antilogarithm  is  1.036812  ...  In  work- 
ing with  four-place  tables,  we  commonly  carry  out  our 
antilogarithms  to  four  (or  sometimes,  in  the  early  part 
of  the  table,  to  five)  significant  figures ;  but  the  last 
figure  must  be  regarded  as  probable,  rather  than  abso- 
lutely certain,  owing  to  the  accumulation  of  small  errors 
in  the  various  logarithms  employed  to  obtain  the  result. 
Thus,  slightly  different  answers  are  sometimes  found  by 
different  methods  of  performing  one  example,  or  even  in 
looking  out  a  single  logarithm  or  antilogarithm  by  differ- 
ent processes  of  interpolation.  This  ought  not  to  pro- 
duce any  distrust  of  the  method  of  logarithms,  but  rather 


COMPUTATION   BY    LOGARITHMS.  37 

to  impress  on  the  student  the  fact  that  results  derived 
from  imperfect  data  must  be  subject  to  corresponding 
imperfections.  In  general,  it  may  be  said  that,  in  all 
computations,  whether  logarithms  are  used  or  not,  the 
results  should  not  be  liable  to  such  errors  as  would  be 
perceptible  in  observation ;  nor  is  it  a  sound  method  of 
working  to  attempt  to  carry  them  decidedly  beyond  the 
point  of  accuracy  which  observation  could  detect. 

All  that  has  been  said  in  this  section  must  be  modified 
by  the  remark  that  the  experienced  computer  may  and 
should  permit  himself  a  freedom  in  the  use  of  his  tables 
which  would  be  unsafe  for  a  beginner.  But  good  habits 
of  approximate  work  are  only  acquired  by  those  who 
have  first  learned  to  be  exact. 

Table  of  Proportional  Parts. 

§46.  This  table  is  designed  to  be  used  in  connex- 
ion with  other  tables,  as  an  aid  in  interpolation.  It  con- 
tains the  product  of  every  integer  from  1  to  100  by  every 
tenth  from  0.1  to  0.9 ;  and  is  readily  used  for  multiplying 
any  number  of  two  figures  by  any  decimal.  The  table  is 
arranged  in  five  rows  of  twenty  columns  each.  The  value 
of  the  multiplicand  is  printed  in  full-faced  type  over  every 
fifth  column,  and  the  values  of  the  multiplier  are  printed, 
without  the  decimal-point,  on  the  right  and  left  of  each 
row  of  columns.  The  products  are  given  as  if  the  mul- 
tipliers were  tenths.  Thus  0.7  X  28  =  19.6,  and  this 
result  is  found  in  the  column  of  28  and  the  line  of  7. 
If  the  multiplier  is  in  the  place  of  hundredths,  thou- 
sandths, or  any  other  numeral  place,  it  is  only  necessary  to 
make  a  change  in  the  place  of  the  decimal-point  in  the 
product.  Thus  0.03  X  62  =  1.86.  To  multiply  a  number 
of  two  figures  by  any  decimal,  we  must  find  the  products 
which  correspond  to  the  successive  figures  of  the  multi- 
plier and  add  them  together.  The  decimal  part  of  the 
4 


38  THE   ELEMENTS    OF   LOGARITHMS. 

result  is  generally  to  be  discarded,  and  in  that  ease  the 
rule  of  §  40  must  be  observed.  Thus,  let  it  be  required 
to  find  0.619  X  37.     We  have 

0.6      X  37=22.2 
0.01    X  37=   0.37 
0.009  X  37=   0.333 
.-.  0.619  X  37  =  23. 

In  like  manner,  we  find  0.27  X  15  =  4,  0.59  X  73  = 
43,  0.78  X  69  =  54,  0.96  X  84  =  81,  0.36  X  57  =  21, 
0.289  X  51  =  15,    0.483  X  93  =  45,    0.374  X  82  =  31. 

The  table  can  also  be  used  inversely.  Thus,  let  it  be 
required  to  find,  to  two  places  of  decimals,  what  part  36 
is  of  79.  Looking  in  the  column  of  79,  we  find  31.6  = 
0.4  X  79  ;  36  —  31.6  =  4.4,  and  the  tabulated  product 
nearest  to  this  is  4.74  =  0.06  X  79. 

.-.ft  =  0.46. 

In  like  manner,  we  find  ff  =  0.43,  |f  =  0.81,  §£  = 
0.32,  f f  =  0.37,  ff  =  0.79,  11  =  0.30. 

The  student  should  practise  himself  in  the  use  of  this 
table  till  he  can  obtain  such  results  as  those  found  in  this 
section  mentally  and  by  a  rapid  glance. 

Three-Place  Table  of  Logarithms  of  Numbers. 

§  47.  This  table  contains,  to  three  places  of  deci- 
mals, the  mantissa  of  the  denary  logarithm  of  every  num- 
ber of  one  or  two  significant  figures.  The  antilogarithm 
is  the  argument  of  the  table,  and  is  printed  in  full-faced 
type,  the  first  significant  figure  being  given  in  the  left- 
hand  column,  and  the  second  significant  figure  (for  which 
0  is  to  be  substituted,  if  there  is  but  one  significant  figure) 
at  the  top  of  the  table.  The  mantissa  of  the  logarithm  is 
given,  in  common  type,  in  the  line  and  column  deter- 
mined by  the  two  figures  of  the  antilogarithm.      Thus, 


COMPUTATION  BY  LOGARITHMS.  39 

we  find  log  8.7  =  0.940,  log  7  =  0.845,  log  0.61  =  9.785 
—  10. 

The  last  line  of  the  table  contains  the  logarithms  of 
numbers  of  three  figures,  beginning  with  10.  Thus,  we 
find  log  1070  =  3.029. 

The  three-place  logarithm  of  any  number  whatever 
and  the  antilogarithm  of  any  three-place  logarithm  can 
be  found  from  this  table  by  the  method  explained  and 
exemplified  in  §§  38-42,  and  the  process  may  be  shortened 
by  the  use  of  the  table  of  Proportional  Parts.  The  anti- 
logarithm  should  be  computed  to  one  figure  only  beyond 
those  tabulated,  as  it  cannot  be  determined  more  accu- 
rately by  its  three-place  logarithm.  It  must  be  observed 
that,  in  their  order  of  succession,  the  logarithms  run 
across  each  line  from  left  to  right,  and  then  begin  again 
at  the  left  of  the  next  line. 

Let  the  student  find  the  following  logarithms  and  anti- 
logarithms  by  this  table  :  — 


437.2   =2.640, 

log  0.694     =  9.841  - 

-10, 

2.78     =  0.444, 

log  17.92     =  1.253, 

32910.  =  4.518, 

log  0.00217  =  7.336- 

-10; 

log-1  2.335  =  216,  log-1  0.794  =  6.23, 

log-1 1.872  =  74.5,  log-1  (9.647  —  10)  =  0.444, 

log-  *  (8.427  — 10)  =  0.0268,  log-  *  2.533  =  342. 


Four-Place  Table  of  Logarithms  of  Numbers. 

§  4§.  The  arrangement  of  this  table  is  similar  to  that 
of  the  three-place  table.  The  antilogarithm  is  the  argu- 
ment, and  is  given  to  three  significant  figures,  for  the 
last  one  or  two  of  which  0  is  to  be  substituted  in  the  case 
of  a  number  of  less  than  three  figures.  The  first  two 
significant  figures  must  be  sought  at  the  left  of  the 
table,  in  the   column   headed  "Natural   Numbers,"  and 


40  THE    ELEMENTS    OF    LOGARITHMS. 

the  third  at  the  top  of  the  table.  The  second,  third,  and 
fourth  decimal-figures  of  the  logarithm  will  be  found  in 
the  line  and  column  thus  determined,  and  the  first  deci- 
mal-figure in  the  0  column,  and  either  on  the  same  line 
with  the  first  two  significant  figures  of  the  antilogarithm, 
or,  if  no  figure  stands  there  in  the  first  decimal  place, 
then  on  the  nearest  line  above  which  contains  a  figure  in 
that  place.  Thus,  log  3.23  =  0.5092,  log  48.6  =  1.6866, 
log  7480=3.8739,  log  0.285=9.4548  —  10,  log  921  = 
2.9643,  log  0.0052  =  7.7160  —  10. 

If  the  second  decimal-figure  is  printed  in  small  type, 
the  first  decimal-figure  is  to  be  sought  at  the  beginning 
of  the  next  following  line.  Thus,  log  2.59  =  0.4133,  log 
126  =  2.1004,  log  0.0797  =  8.9015  — 10,  log  63.2  =  1.8007, 
log  50400  =  4.7024. 

As  in  the  three-place  table,  the  order  of  succession  of 
the  logarithms  is  from  left  to  right  through  the  succes- 
sive lines ;  but  there  are  no  repetitions  in  this  table,  and 
the  last  logarithm  in  each  line  is  followed  by  the  first  in 
the  next  line. 

The  four-place  logarithm  of  any  number  or  the  anti- 
logarithm  of  any  four-place  logarithm  can  be  found  from 
this  table  by  the  method  of  §§  38-42,  by  the  aid  of  the 
table  of  Proportional  Parts,  already  explained.  The  anti- 
logarithm  should  generally  be  computed  to  four  figures 
only,  that  is,  to  one  figure  beyond  the  tabulated  figures ; 
but  in  the  early  part  of  the  table  it  may  be  computed  to 
five  figures.     Thus,  we  have 

log  59.43       =  1.7740,  log  0.008147  =  7.9110  — 10, 

log  284.8       =2.4545,  log  572610.    =5.7579, 

log  0.073748  =  8.8678  —  10,  log  0.5017  =  9.7004  — 10, 
log  3.1607      =  0.4998,  log  99.968     =  1.9999 ; 

log-1  1.9155  =  82.32,   log-1  (5.8760  —  10)  =  0.00007517, 
log-1  0.0547  =  1.1342,  log-1  (8.1577  —  10)  =  0.014377, 
log-1  3.8291  =  6747,    log-1  (9.5757  —  10)  =  0.3765. 


COMPUTATION   BY    LOGARITHMS.  41 

§  49.  Interpolation  in  this  table  is  facilitated  by  the 
use  of  the  right-hand  division  of  the  table,  headed  "  Pro- 
portional Parts."  If  we  follow  out  the  line  which 
contains  the  tabulated  logarithm  corresponding  to  the 
first  three  significant  figures  of  the  given  number,  we 
shall  find,  under  the  fourth  significant  figure,  the  correc- 
tion which  corresponds  to  that  figure,  pointed  off  as  if 
the  mantissas  in  the  table  were  whole  numbers,  and,  in 
like  manner,  under  the  fifth  significant  figure  of  the  anti- 
logarithm,  we  shall  find  the  correction  for  that  figure,  pro- 
vided we  move  the  point  in  that  correction  one  place  to 
the  left,  and  so  on.  Thus,  suppose  it  is  required  to  find 
log  1273.64.    We  find 

mant  log  127  =  1038,  correction  for  .3     ==  10.4 

„         „   .06   =   2.09 
„         „   .004=   0.139 


„   .364  =  13 
/.log  1273.64  =  3.1051. 

In  like  manner,  we  find 

log  0.073748  =  8.8678  — 10,  log  0.287428  =  9.4586  —10, 
log  3.1607     =  0.4998,  log  10.0086   =  1.0004, 

log  908.4       =2.9583,  log  0.09994    =8.9998—10. 

In  using  this  table  of  Proportional  Parts,  we  should 
work  from  the  nearest  tabulated  value.  For  each  line  of 
corrections  is  computed  from  the  average  value  of  the 
tabular  difference  for  that  line,  and  the  smaller  the  pro- 
portional part  taken,  the  less  is  the  probability  of  error. 
Hence,  if  the  fourth  significant  figure  is  as  great  as  5,  we 
should  increase  the  third  significant  figure  by  1,  and  from 
the  tabulated  logarithm  thus  obtained  subtract  a  suitable 
correction.  For  example,  suppose  it  is  required  to  find 
log  1.2084.    We  have  120.84  =  121  —  0.16.    Then 


42  THE   ELEMENTS    OF   LOGABITHMS. 

mant  log  121  =  0828,  correction  for  0.1    =  3d 

„   0.06  =  2.09 
*   0.16  =  6 
.-.  log  1.2084  =  0.0822 
In  like  manner,  we  find 

log  0.3077  =  9.4882  — 10,  log  63.071  ==  1.7998, 
log  5518.3  se  3.7418,  log  9.426   =  0.9743. 

§  50 o  This  sub-table  of  Proportional  Parts  can  also 
be  used  inversely.  Thus,  suppose  it  is  required  to  find 
log-1  1.9155.     We  have 

9154  ==  mant  log  823,   9155  =  9154  +  1, 
1.1  =  correction  for  0.2 ; 
.-.  log-1  1.9155  =  82.32. 

Again,  if  log-1  3.8291  is  required,  we  have 

8293  =j  mant  log  675,   8291  =  8293  —  2, 

1.9  =  correction  for  0.3,  675  —  0.3  =  674.7 

.-.  log-i  3.8291  =  6747. 

In  like  manner,  we  find 

log-1  0.4438  ==  2.779,    log-1  (5.8760  —  10)  =  0.00007516, 
log-1 0.0547  =  1.1342,  log-1  (8.1577  —  10)  =  0.014377, 
log-1 2.8272  =  671.7,    log-1  (9.5757  — 10)  ==  0.3764. 

§  51.  The  extreme  left-hand  column  of  the  table  of 
logarithms,  headed  "Angles,"  is  introduced  to  facilitate 
finding  the  logarithm  of  the  number  of  minutes  in  an 
angle  expressed  in  degrees  and  minutes  or  hours  and 
minutes,  or  of  seconds  in  an  angle  expressed  in  minutes 
and  seconds.  The  unit-figure  is  to  be  taken  at  the  top  of 
the  table,  and  the  characteristic  is  always  2.  Thus  we 
obtain  log  (6°  27')  =  2.5877,  log  (10*  12»)  =  2.7868, 
log  (4'  08")  =  2.3945,  log  (13°  15'.8)  ==  2.9008. 

The  small  table  under  the  table  of  logarithms  is  to  be 


COMPUTATION   BY    LOGARITHMS.  43 

used  for  trigonometric  work,  and  will  be  explained  in 
another  place. 

Arithmetical  Complement.    Reciprocal. 

§  452.  The  arithmetical  complement  of  the  logarithm 
of  a  number,  which  may  also  be  called  the  cologarithm 
of  the  number,  is  the  remainder  obtained  by  subtracting 
that  logarithm  from  10.  It  is  commonly  denoted  by 
"arco  log"  or  by  "  colog."  The  above  definition  is  then 
expressed  by  the  formula, 

colog  x  =z  10  —  log  x.  (33) 

Since 

10  =  9.99 9  -f-  (1  in  the  last  place), 

the  arithmetical  complement  may  be  found  by  subtracting 
each  figure  of  the  logarithm  from  9,  and  then,  adding  1  to 
the  last  figure  of  the  remainder,  or,  in  other  words,  by 
beginning  with  the  characteristic  and  subtracting  each  suc- 
cessive figure  of  the  logarithm  from  9,  down  to  the  last  sig- 
nificant figure,  and  subtracting  that  figure  from  10.  By 
this  method  the  cologarithm  of  a  number  can  be  written 
down  from  inspection  of  the  table  almost  as  rapidly  as 
the  logarithm  itself.     Thus  we  have 


'&■ 


colog  59.43       =   8.2260,  colog  0.008147  =  12.0890, 

colog  284.8       =   7.5455,  colog  572610.   =  4.2421, 

colog  0.073748  =  11.1322,  colog  0.5017      =  10.2996, 

colog  3.1607     =   9.5002,  colog  99.968     =   8.0001, 

colog  1273.64  =   6.8949,  colog  908.4       =   7.0417, 

colog  0.287428  =  10.5414,  colog  10.0086  =   8.9996, 

colog  1.2084     =  9.9178,  colog  0.3077     =10.5118. 

§  53.  By  (33)  and  (9), 

colog  x  — 10  =  —  log  x  =  log  - ;  (34) 


44 


THE    ELEMENTS    OP    LOGARITHMS. 


that  is,  if  10  be  subtracted  from  the  cologarithm  of  any 
number,  the  remainder  is  the  logarithm  of  the  reciprocal 
of  that  number.    Thus  we  have 


z^s 

=  0.01683, 

tt.ooVttt 

=  122.74, 

irsVs 

=  0.003512, 

Z72^TJ) 

=  0.000001746, 

t.tttW** 

=  13.560, 

^■sVtT 

=  1.993, 

^•tV(J7 

=  0.3164, 

■^•^.V^-g" 

=  0.010002, 

t^tVs? 

=  0.0007850, 

¥oW 

=  0.0011008, 

tf.TsV 4  2""S 

=  3.478, 

TU.fos'S 

=  0.09990, 

T.2oS¥ 

=  0.8276, 

TJ.^OTT 

=  3.249. 

Multiplication  by  Logarithms. 

§  5J:«  Multiplication,  division,  involution,  and  evo- 
lution are  easily  performed  by  logarithms  by  the  aid  of 
the  theorems  of  §  16.  The  theorem  for  multiplication  is 
that  the  logarithm  of  the  continued  product  of  several 
quantities  is  the  sum  of  their  logarithms  ;  or 

log  TIx  =  Zlog  x. 

If  any  of  the  factors  are  less  than  1,  their  logarithms 
should  be  increased  by  10,  as  in  §  30,  and  the  resulting 
logarithm  diminished  by  10  times  the  number  of  such 
factors. 

If  any  of  the  factors  are  negative,  their  logarithms  are 
to  be  taken  without  regard  to  their  signs,  and  the  sign  of 
the  result  determined  by  the  rules  of  algebra.  It  is  well 
to  write  the  letter  n  after  any  logarithm  which  corresponds 
to  a  negative  number.  Then  if  the  number  of  logarithms 
so  marked  is  odd,  the  product  is  negative ;  if  even,  the 
product  is  positive.  This  indicates  the  usual  method  of 
dealing  with  negative  numbers  in  logarithmic  compu- 
tation. 

Ex.  1.  Find  908.4  X  (—0.79)  X  (—101.92)  X  3.1607 
x  (—0.0004). 


COMPUTATION  BY   LOGARITHMS.  45 


10 


Am 


908.4 

log 

2.9583 

—  0.79 

log 

9.8976?* 

—  101.92 

log 

2.0083;*, 

3.1607 

log 

0.4998 

—  0.0004 

log 
log 

6.6021n 

—  92.50 

1.9661^ 

10 


Ex.  2.  Find  0.05392  X  0.9  X  1534.1  X  2.117. 

Arts.  157.6. 

Ex.  S.  Find  (—7.1309)  X  0.004006  x  0.7172  x 
(—0.015)  X  21.362.  A)is.  0.006564. 

Division  by  Logarithms. 

§  55,  The  logarithm  of  a  quotient  is  found  by  sub- 
tracting the  logarithm  of  the  divisor  from  that  of  the 
dividend.    By  this  principle  and  by  (34), 

x 
log  —  =  log  x  —  log  y  =  log  x  -\-  colog  y  — 10.    (35) 

if 

Hence  the  logarithm  of  a  quotient  may  be  found  by  add- 
ing together  the  logarithm  of  the  dividend  and  the  cologa- 
rithm  of  the  divisor  and  subtracting  10  from  the  residt. 
The  student  should  habituate  himself  to  working  by  this 
latter  rule,  as  it  is  more  convenient  than  the  former,  espe- 
cially where  compound  operations  are  to  be  performed,  as 
in  §§  56  and  59. 

If  the  characteristic  of  the  dividend  is  increased  by  10, A 
that  of  the  quotient  must  be  diminished  by  20. 
Ex.  1.  Divide  0.9316  by  15.42. 

0.9316  log        9.9692  —  10 

15.42        colog        8.8119 
Ans.        0.06041         log        8.7811  —  10 

Ex.  2.  Divide  0.039008  by  —  0.0097.    Ans.  —  4.022. 

Ex.  3.  Divide  —  67.18  by  —  0.4573.     Ans.  146.93. 


46  THE   ELEMENTS    OF   LOGARITHMS. 

The  Rule  of  Three  by  Logarithms. 

§  56.  If  any  three  terms  of  a  proportion  are  known, 
the  fourth  can  be  found  by  the  Rule  of  Three ;  and  the 
operations  required  by  this  rule  may  be  performed,  in  a 
single  process,  by  logarithms.     See  §  17. 

Ex.  1.  0.5334  :  x  —  —0.001907  :  —2.008.    Find  x, 

rru    j>  i      *  rru         •               °-5334  X  (—2.008) 
The  Rule  of  Three  gives  x  = _0t001907 — - 

0.5334  log         9.7270  —  10 

—  2.008  log         0.3028» 

—  0.001907  colog  12.7197^ 
x  =  561.7  log         2.7495 

Ex.  2.  —37.82  :  503.7  =  0.09852  :  x.    Find  x. 

Ans.  x  =  — 1.3119. 

Ex.  S.  0.71972  :  3.156  =  x  :  0.04983.    Find  as. 

Ana.  x  =  0.011363. 

Involution  by  Logarithms. 

§  57,  The  logarithm  of  any  power  of  a  quantity  is 
found  by  multiplying  the  logarithm  of  the  quantity  by 
the  exponent  of  the  power  /  or 

log  xn  =  n  log  x. 

If  the  logarithm  of  the  given  quantity  is  increased  by 
10,  the  product  must  be  diminished  by  10  times  the  expo- 
nent of  the  required  power,  since 

n  (log  as-J— 10)  =n  log  x -f-  lOn. 

The  results  found  by  this  method  are  hardly  to  be 
relied  on  beyond  the  third  significant  figure,  since  the 
error  in  the  logarithm  of  the  given  quantity  is  multiplied 
by  the  exponent. 


COMPUTATION    BY    LOGARITHMS.  47 

Ex.  1.  Find  the  cube  of  0.8724 

0.8724        log        9.9407  —  10 
3 


Ans.  0.664  log        9.8221  —  10 

Ex.  2.  Find  the  7th  power  of  3.051.    Ans.  2400. 
Ex.  3.  Find  the  square  of  21.27.    Ans.  452.5. 

Evolution  by  Logarithms. 

§  58.  The  logarithm  of  any  root  of  a  quantity  is 
found  by  dividing  the  logarithm  of  the  quantity  by  the 
exponent  of  the  root ;  or 

n  I 

log  \/x  = log  X. 

n 

If  the  characteristic  of  the  logarithm  of  the  given 
quantity  is  negative,  it  should  be  increased  before  division, 
not  by  10,  but  by  10  times  the  exponent  of  the  required 
root,  and  then  the  characteristic  of  the  quotient  should  be 
decreased  by  10;  for 

log  x  4-  lOrc       log  x   .    .  A       ,       »       .    %  . 

— =  — \-  10  =  log  \fx  -j-  10. 

n  n 

Ex.  1.  Find  the  4th  root  of  0.005627. 

0.005627  4)37.7503  —  40 

Ans.     0.2739  log       9.4376  —  10 

Ex.  2.  Find  the  9th  root  of  0.0007808. 

Ans.  0.4516. 
Ex.  3.  Find  the  5th  root  of  370.42. 

Ans.  3.264. 
Ex.  ^.  Find  the  cube  of  the  11th  root  of  61.98. 

Ans.  3.081. 
Ex.  5.  Find  the  10th  power  of  the  7th  root  of  0.8213. 

Ans.  0.7550. 


48  THE   ELEMENTS    OF   LOGARITHMS. 

Compound  Operations, 

§  50.  Examples  involving  the  combination  of  multi- 
plications, divisions,  involutions,  and  evolutions  are  espe- 
cially suited  to  be  solved  by  logarithms. 

Let  the  student  work  out  the  following  results:  — 

Ex.  1. 


a 

V 


0.05639  X  1.07283  X  4^21.3  X  10.00 

: ■ ■  =  4.068. 


V0.00762310  X  52.46  X  0.7000 
Ex.  2. 


/V7394.  xM2998y        /  532.4  X  0.2900  _ 

V  1.00632  X  57.00    /  X  V  V0.0695  X  9.995  _  u-uuu^uy- 

Ex.  3. 

3U         \/Q.019Q73  X  3.796  y=  g574 

V  \0.05545*  X  1000.0  X  0.0007241/ 

Ex.  4- 

4  /  \^43.04  X  809.8  X  0.25163  X  0.09093     _  0  2386> 

V    2139.  X  3.007  X  ^0.008188  X  0.74334 

Ex.  5. 
"l.00075X  0.3710    .   ^45.56x0.2203 


V      \A).005278         *     7.197  X  0.6411 

Ex.  6. 

/0.7628  X  79328.  X  0.01333%  {%$ 


8.140. 


40.07  X  Vl5.21 


-)     =o. 


2034. 


Exponential  Equations. 
[§  60.  We  have  seen,  in  §  18,  that  if  5*  =  my 

log  rn 

~  log  b * 


COMPUTATION   BY   LOGARITHMS.  49 

or  log  x  =  log  log  m  —  log  log  b  \         ,^q\ 

=  log  log  m  -j-  colog  log  b  —  10.  > 

Ex.  1.  21*  =  15.75.     Find  x. 

m  =  15.75     log  1.1973      log  0.0783 
b    =21         log  1.3222  colog  9.8787 

x   =0.9058  log  9.9570  — 10 

Ex.  2.  40.56*=  8276.    x  =  2.437.] 

[§  61.  If  either  b  or  m  is  less  than  1,  log  b  or  log  m  is 
negative,  and  we  proceed  as  in  §  54. 

Ex.  1.  Find  x  from  the  equation  0.5175*  =  6.208. 

m  =  6.208    log  0.7930  log     9.8993  —  10 

b  =  0.5175  logT.7139  =  — 0.2861  colog  10.5434;* 
x  =  —  2.771.  log     0.4427/1 

Ex.  2.  Find  x  from  the  equation  0.027907*  =  0.8213. 

Ans.  x  =  0.05501.] 

[§  OS.  It  has  already  been  remarked  in  §  19  that  solv- 
ing an  exponential  equation  by  logarithms  is  equivalent 
to  converting  a  logarithm  from  one  system  to  another. 
Thus,  in  the  above  examples,  we  have  found  the  following 
logarithms  from  their  values  in  the  denary  system:  — 

log21    15.75  =  0.9058,    log0.i5175  6.208  =  —  2.771, 
log4o.56  8276  =  2.437,      log/usm  0.8213  =  0.05501.] 

Logarithms  of  Sums  and  Differences. 

[§  63.  Addition  and  Subtraction  cannot  be  performed 
by  logarithms.  Hence  arise  inconvenience  and  liability 
to  error,  when  these  operations  occur  in  the  midst  of 
others  which  can  be  performed  by  logarithms.  The  table 
of  Logarithms  of  Sums  and  Differences  is  one  form  of 
a  table  proposed  by  Gauss  to  obviate  this  difficulty. 


50  THE   ELEMENTS   OF   LOGARITHMS. 

The  argument  of  this  table,  designated  by  the  letter 
A,  is  any  logarithm,  which  we  may  call  log  x.  Its  char- 
acteristic (increased  by  10,  if  negative)  is  at  the  top  of 
the  table,  and  the  first  two  figures  of  the  mantissa  are 
in  the  left-hand  column.  The  remaining  figures  of  the 
mantissa  are  0  for  the  tabulated  values  of  the  argument. 
The  function  given  in  the  table,  designated  by  B,  is  log 
(x  -\-  1).  Its  characteristic,  accompanied  sometimes  by 
one  or  two  figures  of  the  mantissa,  is  printed  at  the  top 
and  bottom  of  each  column.  The  remaining  figures  are 
to  be  found  in  the  body  of  the  table ;  the  first  of  them 
being  printed  only  in  the  first  line  in  which  it  has  a  new 
value  and  also  in  the  first  line  of  each  division  of  five 
lines.  When  this  first  figure  is  in  small  type,  the  earlier 
figures  of  B  are  to  be  taken  from  the  foot  of  the  column ; 
otherwise  they  are  to  be  taken  from  the  top  of  the 
column. 

We  shall  call  B,  or  log  (x-\-  1),  the  Gaussian  of  A,  or 
log  a:,  and  A  the  anti-  Gaussian  of  B;  and  Ave  shall  use 
the  symbols  (s)  and  ©— 1  to  designate  these  relations. 
Thus  we  write 

B  =  ©A,      or  log  (*  +  1)  ==  ®     log  x,  ) 

A  =  ©-IB,  or  log  (x  —  1)  =  ®-l  log  x\  J  K     } 

where  the  exponent  — 1  is  used  according  to  the   prin- 
ciple laid  down  in  §  34. 
We  have  then 

©  (9.2100  —  10)  =  0.0653,     ©   0.9800  =  1.0232, 

®    0.6300  =  0.7215,     ©  (8.4800  —  10)  =  0.0129. 

These  results  may  be  verified  by  means  of  the  table 
of  Logarithms  of  Numbers ;  which  gives,  for  instance, 

log-1  (9.2100  —  10)  =  0.1622  =  a-, 
log  (x  -f  1)  =  log  1.1622  =  0.0653.] 

[§  64,  The  Gaussian  or  anti-Gaussian  of  any  four-place 


COMPUTATION   BY   LOGARITHMS.  51 

logarithm  may  be  found  to  four  decimal-places  from  this 
table  by  simple  interpolation.  In  the  principal  columns 
of  the  table,  a  multiplier  (derivative),  to  be  used  instead 
of  the  tabular  difference,  is  printed  in  small  type  after 
each  value  of  the  function.  This  multiplier  should  prop- 
erly be  used  through  half  the  tabular  interval  on  each 
side  of  the  value  to  which  it  is  attached ;  or,  in  other 
words,  we  should  apply  our  correction  to  the  nearest 
tabulated  value.  The  general  table  of  Proportional  Parts 
may  be  used  in  computing  corrections. 

The  multiplier  is  omitted  in  the  earlier  columns  of  the 
table,  where  the  tabular  differences  are  so  small  that  cor- 
rections are  easily  computed,  and  also  in  the  later  columns, 
after  the  occurrence  of  the  letter  c,  where  its  value  is  100, 
and  A  and  B  vary  by  equal  amounts. 

The  table  extends  only  from  A  =  6.0000  —  10  to  A  = 
4.0000.     The  reason  of  this  limitation  is  that 

if  A  <  6.0000  —  10,  B  =  0.0000  to  four  places, 
and  if  A  >  40000,  B  =  A  to  four  places. 

Let  the  student  find  the  following  values  from  this 
table  :  — 

©   1.0960  ==1.1295,      ©   3.8129  =3.8130, 

©  (7.5265  —  10)  =  0.0015,      ©  (9.6431  —  10)  ==  0.1582. 

©-11.0960  =  1.0597,  ©-13.8129  =  3.8128, 

©-10.1051  =  9.4373  —  10,        ©-11.0216  =  0.9782.] 

[§  G«>o  If  m  and  n  are  any  two  numbers,  we  have 

m-\-n=zn  (—  +  l)»    m  —  n=n( 1), 

log  (m  +  n)  =  log  n  -f-  log  (^  +  1 V 

log  (m  —  n)  =  log  n  +  log  C^-  — 1\  ; 


52  TIIE    ELEMENTS    OF    LOGARITHMS. 

or,  by  (37), 

771 

log  O |«)=logw  +  @log- 

=  log  n-\-($  (log  m  +  colog  ??)> 
log  (>?i  —  n)  =  log  »  -f-  ©-1  log  — 

±=  log  n  -f-  G)-1  (log  «i  +  colog  n)  ; 


(38) 


and  by  these  formulas  we  can  obtain  log  (m-f-n)  and  log 
(m  —  w)  when  log  m  and  log  w  are  known,  without  having 
to  find  in  and  ?i  themselves. 

Putting  in  (38)  ?w  =  l  and  ?i  =  x,  we  have,  by  (7), 

log  (1  +  sb)  =  log  x  +  ®  colog  aj,  j 

log  (1  —  #)  =  log  x  -f-  ©-1  (colog  x)    )  ^     ' 

Formulas  (37-39)  will  be  found  at  the  bottom  of  the 
table.] 


[§  66.  The  use  of  this  table  is  illustrated  by  the  fol- 
lowing examples :  — 

Ex.  1.  Find  the  hypotenuse  h  of  a  right  triangle  of 
which  the  legs  are  a  =  43.76  and  b  =  71.02. 

The  Pythagorean  Proposition  gives  h2  =  a2  -f-  b2  ,  or, 
h  =  \/(a2  -f-  b2).  Working  without  the  Gaussian  table, 
we  have 

a  =  43.76    log  1.6411         b  =  71.02    log  1.8514 
2  2 


«2  =  1915    log  3.2822  log  3.7028 

b2  =  5044 

a? +0*=  6959      2)3.8425 
h  =83.40  log  1.0212 

The  Gaussian  method  saves   two   references   to    the 
table.     By  that  method  we  have 


COMPUTATION   BY    LOGARITHMS.  53 


log  a    =1.6411 

colog  b     =  8.1486 


log^    =9.7897  —  10 
0  b 

log  ^-  =  9.5794  — 10  ©  0.1397=  log  C*  +  ** 

I  ©  0.0698  =  log         1  — 

log  b  1.8514 
h  =  83.40       log     1.9212 
-Eb.  2.  Compute 

(0.2194)2  x  V^915.7  +  4.4  X  V'O/Tx  32.03 
(571.6  X  0.0235  —  0.04004  X  6.72  X  10.97)3 

Am.  0.1143.] 


APPENDIX. 


EXPLANATION  OF  THE  TRIGONOMETRIC  TABLES. 


(§  1.)  The  following  formulas,  in  which  qp  denotes  any 
angle,  are  proved  in  Trigonometry :  — 


log  sin  cp  ==  colog  esc  qp  —  10, 
log  esc  qp  =  colog  sin  qp  —  10, 
log  tan  qp  =  colog  ctn  qp  —  10, 
log  ctn  qp  =  colog  tan  qp  —  10, 
log  sec  qp  =  colog  cos  qp  —  10, 
log  cos  cp  =  colog  sec  qp  —  10.  t 


(A) 


See  Seaver's  Formulas,  6 ;  Peirce's  Trigonometry,  §  10 ; 
Chauvenet's  Trigonometry,  §  19  ;  Elements  of  Logarithms, 
§53. 


sin  (90°  —  qp)  =  cos  qp, , 
esc  (90°  —  qp)  =  sec  qp, 
tan  (90°  —  g>)  =  ctn  qp, 
ctn  (90°  —  cp)  =  tan  qp, 
sec  (90°  —  qp)  =  esc  qp, 
cos  (90°  —  qp)  =  sin  cp. 


(B) 


See  Seaver,  35 ;  Peirce,  §  7 ;  Chauvenet,  §§  18,  38. 


56  THE    ELEMENTS    OF   LOGARITHMS. 

sin  (90°  -f-  q>)  =       cos  <jp, 
cos  (90°  -|-  op)  =  —  sin  r/>, 
tan  (90°  -f-  <p)  =  —  ctn  cp ; 
sin   cp  =       (— )*  sin  (2k  90°  +  q>) 

=  —(—)*  cos  ([2k  -f  1]  90°  +  cp),    )  (C) 
cos  9  =      (— )*  cos  (2£  90°  +  ?) 

=       (— )*  sm  (PA+l]90o+qp), 
tan  g>  =  tan  (2k  90°  +  q) 

=        —      ctn  ([2&  +  1]  90°  +  cp). 

See  Seaver,  35,  28,  29,  30  ;  Peirce,  §§  63,  65,  68 ;  Chau- 
venet,  §§  38,  41,  43,  45,  51,  52. 


<P) 


1st  Qu.  2nd  Qu.  3rd  Qu.  4th  Qu. 

sin        +  +            —            — 

cos        -f-  —            —            + 

tan        +  —            +            — 

See  Seaver,  4,  5 ;  Peirce,  §§  62,  66 ;  Chauvenet,  §  47. 


Three-Place  Table  of  Trigonometric  Functions. 

(§  2.)  This  table  contains  the  values  of  the  six  simple 
trigonometric  functions,  generally  to  three  significant 
figures,  and  their  logarithms  to  three  places  of  decimals, 
for  every  degree  of  the  quadrant.  The  logarithms  of  the 
functions  occupy  the  left-hand  division  of  the  table,  and 
the  natural  values  the  right-hand  division.  The  degrees 
run  down  the  left-hand  column  of  the  table  to  45°,  and 
then  up  the  right-hand  column.  The  names  of  the  func- 
tions are  to  be  taken  from  the  top  of  the  page  when  the 
degrees  are  taken  from  the  left-hand  column,  marked  °  at 
the  top,  and  from  the  bottom  of  the  page  when  the  de- 
grees are  taken  from  the  right-hand  column  marked  °  at 
the  bottom.  The  integral  part  of  the  value  of  the  func- 
tion, whether  natural  or  logarithmic,  is  generally  printed 
only  at  the  top  and   bottom   of  the  column,  unless  it 


APPENDIX.  57 

changes  its  value  in  the  column.  The  characteristics, 
when  negative,  are  increased  by  10,  so  as  to  become  posi- 
tive. This  is  the  case  with  the  log  sin  and  log  cos  of  all 
angles  and  with  the  log  tan  of  angles  less  than  45°  and 
the  log  ctn  of  angles  greater  than  45°.  We  shall  find  it 
convenient  to  follow  this  usage  in  citing  the  logarithmic 
functions,  in  our  illustrations,  and  to  leave  the  term  —  10 
to  be  added  (or  remembered)  by  the  student  when  he  em- 
ploys the  functions  in  examples.     We  have  then 

sin  17°  =  0.292,       tan  17°  =  0.306,      sec  17°  =  1.046, 
esc  17°  =  3.42  ctnl7°  =  3.27,         cosl7°  =  0.956; 

Iogsinl70  =  9.466,logtanl7°  =  9.485,logsecl7°  =  0.019, 
Iogcscl7°  =  0.534,logctnl70  =  0.515,logcosl7°  =  9.981; 

sin  52°  =  0.788,       tan  52°  =  1.28,         sec  52°  =  1.62, 
esc  52°  =  1.269,       ctn  52°  =  0.781,       cos  52°  =  0.616; 

Iogsin52°  =  9.897,logtan520  =  0.107,logsec52°  =  0.211, 
log  esc  52°  =  0.103,  log  ctn  52°  =  9.893,  log  cos  52°  =  9.789. 

The  method  of  simple  interpolation  explained  in  the 
treatise  on  Logarithms,  §§  38-42,  may  be  applied  to  this 
table  both  directly  and  inversely.  The  tabular  differences 
are  printed  between  the  lines  in  small  type,  except  for  the 
last  two  columns  of  logarithmic  functions.  The  table  of 
Proportional  Parts  may  be  used  in  computing  the  correc- 
tions. The  student  must  be  careful  to  obseiwe  whether 
the  correction,  in  any  given  case,  should  be  applied  to 
the  tabulated  value  by  addition  or  subtraction.  This  can 
be  determined  by  the  theorem  that  the  sine,  tangent,  and 
secant  of  an  acute  angle  increase  with  the  increase  of  the 
angle,  while  the  cosine,  cotangent,  and  cosecant  decrease; 
or,  more  simply,  by  the  principle  that  corresponding 
values  of  the  argument  and  the  function  must  lie  within 
corresponding  tabular  differences.  For  example,  to  find 
the  functions  of  71°. 8,  we  apply  to  those  of  72°  two- 
tenths,  or  to  those  of  71°  eight-tenths,  of  the  tabular  dif- 


58  THE  ELEMENTS  OF  LOGARITHMS. 

ferences  for  the  interval  between  71°  and  72°,  adding  or 
subtracting  in  each  case  so  as  to  bring  the  result  between 
the  tabulated  values  for  71°  and  72°.     Thus  we  obtain 

sin  71°.8  =  0.94G  -f  0.004  =  0.951  —  0.001  =  0.950, 

esc  71°.8  =  1.058  —  0.006  =  1.051  -f-  0.001  =  1.052, 

tan  71°.8  =  2.90    +  0.14    =  3.08    —  0.04   =  3.04, 

ctn  71°.8  =  0.344  —  0.015  =  0.325  +  0.004  =  0.329, 

sec  71°.8  =  3.07    +  0.14    =  3.24    —  0.03   =  3.21, 

cos  71°.8  =  0.326  —  0.014  =  0.309  -f  0.003  =  0.312 ; 

log  sin  71°.8  ss  9.976  -f  0.002  =  9.978  —  0.000  =  9.978, 

log  esc  71°.8  =  0.024  —  0.002  =  0.022  +  0.000  =  0.022, 

log  tan  71°.8  =  0.463  -f  0.020  =  0.488  —  0.005  =  0.483, 

log  ctn  71°.8  =  9.537  —  0.020  =  9.512  +  0.005  =  9.517, 

log  sec  71°.8  =  0.487  +  0.018  =  0.510  —  0.005  =  0.505, 

log  cos  71°.8  =  9.513  —  0.018  =  9.490  +  0.005  =  9.495. 

The  two  methods  of  interpolation  here  indicated  lead  to 
the  same  result.  But  the  best  habit,  as  we  have  remarked 
in  the  book  on  Logarithms,  is  that  of  working  from  the 
nearest  tabulated  value,  which  is,  in  this  case,  72°. 

In  the  inverse  use  of  this  table,  where  an  angle  is  found 
from  the  logarithm  of  one  of  its  trigonometric  functions, 
the  result  should  be  expressed  to  the  nearest  tenth  of  a 
degree.     E.g.,  if 

log  tan  y  =  0.367, 

the  nearest  tabulated  value  of  the  log  tan  is  0.372  =  log 
tan  67°  ;  then,  using  formula  (32)  of  the  book  on  Loga- 
rithms, we  have 

Au  =  21,  y2  -  u  =  5,  ^=p  =  A  =  0.2, 
y  =  67°  —  0.2  =  66°.8. 

In  like  manner,  let  the  student  find  the  following 
values  :  — 

If  log  sin  q.  =  9.325,  cp  =  12°.2;  if  cos  <y  =  0.494,  <p  =  60°.4, 
if  logctn(p  =  9.960,9  =  47o.6;  if  ctn  9=2.05,    cp  =  20°.7. 


APPENDIX. 


59 


(§3.)  It  is  to  be  observed  that  the  two  angles,  taken 
from  opposite  sides  of  the  page,  which  correspond  to  any 
one  number,  found  or  interpolated  in  a  given  column  of 
the  table,  are  complements  of  each  other,  and  also  that  the 
two  names  of  any  column,  given  at  the  top  and  bottom  of 
the  page,  and  corresponding  to  the  two  angles,  are  com- 
plementary.    Thus, 

0.292  =  sin  17°  =  cos  73°,  0.494  =  cos  60°.4  ==  sin  29°.6, 
0.107  =  log  tan  52°  9.633  =  log  tan  23°.2 

=  log  ctn  38°,  =  log  ctn  66°.8. 

This  relation  is  in  accordance  with  formula  (B)  of  (§  1).* 

Again,  it  will  be  seen  that  the  log  sin  and  log  esc,  the 
log  tan  and  log  ctn,  the  log  sec  and  log  cos  are  respectively 
arithmetical  complements.     Thus, 

log  tan  17°  =9.485,  logctnl7°  =  0.515  =  co  9.485; 
log  sec  52°  =0.211,  log  cos  52°  =  9.789  =  co  0.211; 
log  sin  71°.8  =  9.978,   log  esc  71°.8  =  0.022  =  co  9.978. 

This  relation  agrees  with  (§  1)  (A),  since  10  should  be  sub- 
tracted from  one  of  the  logarithms  of  each  pair,  as  taken 
from  the  table,  in  order  to  reduce  it  to  true  value.  It  is  a 
consequence  of  this  relation  that  the  tabular  differences 
are  alike  in  the  columns  of  log  sin  and  log  esc,  also  in  the 
columns  of  log  tan  and  log  ctn,  and  lastly  in  the  columns 
of  log  sec  and  log  cos.  The  pairs  of  columns  thus 
related  are  bracketed  together  in  the  logarithmic  part 
of  the  table,  so  that  the  six  columns  are  printed  as 
three  double  columns. 

Other  formulas  of  trigonometry,  such  as 

log  tan  qp  =  log  sin  qp  -\-  log  sec  qp, 

log  cos   qp  =  log  sin  qp  -|-  log  ctn  qp, 

sin2  qp  -|-  cos2  qp  =  1,  sec2  cp  —  tan2  qp  =  1,  &c, 

*  The  numbers  of  the  sections  of  the  Appendix  are  distinguished  from 
those  of  the  book  by  being  included  in  parentheses. 


60  THE    ELEMENTS    OF   LOGARITHMS. 

are  readily  verified  by  this  table.     Thus  we  have 

log  sin  78°  =  9.990 
log  sec  78°  =  0.682 

log  sin  78°  -f  log  sec  78°  =  0.672 
log  tan  78°  =  0.673; 

where  a  slight  discrepancy  arises  from  the  accumulation 

of  errors  in  the  addition  of  the  logarithms. 

Again, 

log  tan   22°  =  9.606 

log  tan2  22°  =  9.212 
©  log  tan2  22°  =    log  (1  -f  tan2  22°)  =  0.066 
.-.  log  sec   22^  =  i  log  (1  -J-  tan2  22°)  =0.033. 

(§  4.)  Where  the  printed  tabular  differences  are  omit- 
ted near  the  top  of  a  column,  being  printed  in  the  lower 
part  of  the  same  column,  the  principle  of  interpolation 
cannot  properly  be  applied  to  the  table.  In  this  case,  we 
may  employ  the  small  table  in  the  upper  right-hand  corner 
of  the  page.  The  direct  use  of  this  table  is  to  find  the  log 
sin  or  log  tan  of  a  small  angle,  by  finding  the  logarithm 
of  the  angle  itself,  expressed  in  degrees,  and  adding  to 
that  logarithm  a  logarithm  (S  for  the  log  sin  or  7Tfor  the 
log  tan)  of  which  the  value  is  given  in  the  small  table. 
Thus,  we  have 

log  sin  5°.23  =  0.718  -f  8.241  =  8.959, 
log  tan  5°.23  =  0.718  +  8.243  =  8.961. 

The  log  esc  and  log  ctn,  as  we  have  seen  in  (§  3),  are 
the  arithmetical  complements  of  the  log  sin  and  log  tan. 

log  esc  5°.23  =  1.041,  log  ctn  5°.23  =  1.039. 

The  natural  esc  and  ctn  are  the  cuitilogarithms  of  the 
log  esc  and  log  ctn.     E.g., 

esc  5°.23  =  11.00,  ctn  5°.23  =  10.95. 


APPENDIX.  61 

The  log  cos,  log  ctn,  log  sec,  log  tan,  and  natural  sec 
and  tan  of  an  angle  near  90°  are,  by  (B),  found  by  finding 
the  complementary  functions  of  the  complement  of  that 
angle.    E.g., 

log  cos  83°.81  =  logsin  6°.19  =  0.791 +8.241  =  9.032, 
log  ctn  83°.81  =  9.035,     log  sec  83°.81  =  0.968, 
log  tan  83°.81  =  0.965, 
sec  83°.81=  9.30,         tan  83°.  81  =  9.22. 

This  table  can  also  be  used  inversely.  Thus,  to  find 
the  angle  whose  log  sin  is  9.123.  The  large  table  shows 
that  it  is  between  7°  and  8°.  .-.  S=  8.241  or  8.240. 
9.123  —  8.240  =  0.883  =  log  (7.64  <  7.83).  .-.  8=  8.241. 
9.123  —  8.241  =  0.882  =  log  7.62,  /.  9.123  =  log  sin  7°.62. 
In  like  manner,  we  find 

9.055  =  log  sin    6Q.51,        9.120  =  log  tan    7°.52, 
1.470  =  log  ctn    1°.94,        0.962  =  log  sec  83°.73, 
1.245  =  log  esc   3°.26,        8.127  =  log  ctn  89°.232, 
15.75=        ctn   3°.64,        100.0=         esc    0°.573, 
10.00  =        sec  84°.26,        37.71  =         tan    88°.48. 

This  small  table  may  be  explained  by  reference  to  the 
principle  that  the  sine  and  tangent  of  a  small  angle  are 
nearly  equal  to  the  arc  which  is  subtended  by  the  angle  in 
the  unit-circle.  But  the  circumference  of  the  unit-circle 
is  3.14159  .  .  .  Hence,  the  arc  of  1°  in  that  circle  is 
3-1416-  =  0.01745  .  . ,  of  which  the  three-place  logarithm 
is188.242.     Then 

arc  cp  in  unit-circle    =  (cp  in  degrees)  X  0.01745  .  .  , 
log  (arc  cp  in  unit-circle)  =  log  (cp  in  degrees)  -f-  8.242. 

This  formula  is  exact  for  any  angle  cp,  and  if  qp  is  small,  it 
gives  an  approximate  value  of  log  sin  cp  and  log  tan  cp, 
the  accurate  values  to  three  places  being  found  by  sub- 
stituting for  8.242  the  values  of  S  and  T  given  in  the 
table. 
6 


62  THE    ELEMENTS    OF   LOGARITHMS. 

(§5.)  This  table  is  directly  applicable  to  acute  angles 
only.  If  we  have  to  find  a  function  of  an  angle 
in  any  other  quadrant  than  the  first,  we  must  subtract 
from  the  angle  the  greatest  multiple  of  90°  which  it 
contains,  and  then  find  the  same  function  of  the  re- 
mainder if  we  have  subtracted  an  even  multiple,  or  the 
complementary  function  if  we  have  subtracted  an  odd 
multiple.  See  formula  (C).  The  sign  of  the  result  is 
most  readily  determined  by  observing  the  quadrant  in 
which  the  given  angle  lies.  See  (D).  The  sign  affecting 
the  angle  may  be  disregarded  in  the  first  part  of  the  work, 
but  must  be  considered  in  determining  the  quadrant. 
Thus,  if  log  sin  302°.7  is  required,  we  subtract  270^  =  3 
X  90°,  obtaining  32°.7  as  the  remainder.  Then,  as  we 
have  subtracted  an  odd  multiple, 

log  sin  302°.7  =  log  cos  32°.7  =  9.925,  (neglecting  signs). 

But  302°. 7  being  in  the  fourth  quadrant,  where  the  sine  is 
negative, 

log  sin  302°.7  =  9.925?i. 

In  like  manner  we  have 

cos        143°.4   —  —  0.803,  log  tan       205°.6    =9.680, 
sin  (— 1055°.7)  =       0.412,  log  sec  (— 49P.8)  =  0.176ft, 
ctn     (— 16°.1)=  — 3.47,     log  esc        120°.2   =0.063. 

In  using  the  table  inversely  for  angles  in  any  quadrant, 
we  should  observe  that  any  one  value  of  a  trigonometric 
function  corresponds  to  two  angles  between  0°  and  360°,  or 
between  — 180°  and  -f- 180°,  or  in  any  continuous  angular 
space  of  four  right  angles.     Thus, 

if  log  ctn  gp  =  9.663n,  g>=114°.7  or  =  294°.7; 
if  log  sin  y  =  9.886,  y  =  50°.3  or  =  129°.7 ; 
if        cos  cp  =  0.810,     cp  =    3&°.9  or  =  324°.l ; 

if  log  sec  cp  =  0.30b*,  q  =  244°.2  or  =  115°.8. 


APPENDIX.  63 


Four-Place  Table  of  Logarithmic  Trigonometric 
Functions. 

0  6.)  This  table  occupies  five  pages,  the  last  four  of 
which  we  shall  consider  first.  These  pages  contain  the 
logarithms  of  the  sine,  tangent,  cotangent,  and  cosine,  at 
intervals  of  10',  from  5°  to  85° ;  the  characteristics,  when 
negative,  being  increased  by  10.  The  log  esc  and  log  sec 
are,  by  (A)  and  (§  3),  the  arithmetical  complements  of  the 
log  sin  and  log  cos ;  while  the  log  tan  and  log  ctn  are 
arithmetical  complements  to  each  other.  To  find  the 
logarithm  of  any  function  of  any  tabulated  angle,  we  seek 
tbe  degrees,  if  below  45°,  in  the  left-hand  column,  and  if 
equal  to  or  above  45°,  in  the  right-hand  column,  and  take 
the  name  of  the  function  on  the  same  side  of  the  table 
as  the  degrees,  and  immediately  against  the  degrees.  In 
the  line  determined  by  the  name  of  the  function,  and  in 
the  column  determined  by  the  minutes,  taken  at  the  top 
of  the  table  if  the  degrees  are  found  at  the  left,  and  at 
the  bottom  of  the  table  if  the  degrees  are  found  at  the 
right,  will  be  found  the  second,  third,  and  fourth  decimal 
figures  of  the  function.  The  characteristic  and  first  deci- 
mal-figure will  be  found  in  the  left-hand  column  of  the 
values  of  the  function,  unless  the  second  decimal-figure  is 
printed  in  small  type,  and,  in  that  case,  they  are  in  the 
right-hand  column  of  the  values  of  the  function.     E.g., 

log  tan    8P  40'  =  9.1831,  log  sec  38°  00'  =  0.1035, 

log  sin  18°  10'  =  9.4939,  log  esc  52°  00'  =  0.1035, 

log  sin  18°  30'  =  9.5015,  log  sec  59°  50'  =  0.2988, 

log  ctn  32°  20'  =  0.1986,  log  ctn  68°  20'  =  9.5991. 

(§  7.)  The  functions  of  angles  not  tabulated,  and  lying 
between  5°  and  85°,  can  be  found  from  this  table  by  inter- 
polation, care  being  taken  to  apply  the  correction  so  as  to 
bring  the  result  within  the  proper  tabular  difference.     To 


64  THE    ELEMENTS    OF   LOGARITHMS. 

facilitate  the  interpolation,  the  proportional  parts  of  the 
tabular  differences  (except  those  less  than  10)  occurring 
on  the  second  and  third  pages  of  the  table  (pp.  8  and  9) 
are  printed  on  those  pages;  while  the  fourth  and  fifth 
pages  (pp.  10  and  11)  contain  the  corrections  for  each 
line  arranged  as  in  the  table  of  Logarithms  of  Numbers. 
The  latter  plan  is  inapplicable  to  pp.  8  and  9,  because  the 
tabular  differences  vary  so  fast  that  an  average  value  can- 
not safely  be  used  throughout  a  line. 

Thus,  let  the  functions  of  7°  18'.4  (=  1Q  20'  —  1'.6)  be 
required.     We  have 

log  sin         log  tan         log  ctn        log  cos 
for  7°  20'       9.1060,        9.1096,        0.8904,        9.9964, 
tabdiff  99,  101,  101,  2; 

and  the  proportional  parts  of  the  tabular  differences  are 
for  1'  9.9  10.1 

for  0'.6  ^94  (M)6  while  — ^  =  0.32. 

for  1.6  16.  16. 

Then,  applying  the  corrections  so  as  to  bring  the  results 
between  the  values  for  7°  10'  and  7°  20',  we  have 


log  sin  =  9.1044,  log  tan  =  9.1080,  log  cos  =  9.9964, 
log  esc  =  0.8956,  log  ctnr=  0.8920,  log  sec  ==  0.0036. 

Again,  if  the  functions  of  53°  47'.9  (=  53°  50'  —  2'.1) 
are  required,  we  have 

log  sin         log  tan        log  ctn        log  cos 
for  53°  50'     9.9070        0.1361,        9.8639,        9.7710, 

and  the  lines  of  Proportional  Parts  give 

for  log  sin  for  log  tan  and  ctn  for  log  cos 

for  2'                  1.9                        5.3  3.4 

forOU              ^09                      0.26  0.17 

for2U              2.                         6.  4T~ 


APPENDIX.  C5 

Applying  these  corrections,  we  have 

for  53°  47/.0, 

log  sin  =  9.9068,  log  tan  =  0.1355,  log  cos  =  9.7714, 
log  csc=  0.0932,  log  ctn=  9.8645,  log  sec  =  0.2286. 

In  like  manner,  we  find  :  — 

log  tan  12°  41'.3  =  9.3525,  log  cos  37°  28'.7  =  9.8996, 
log  sin  18°  26'.2  =  9.5001,  log  ctn  51°  43'.7  =  9.8970, 
log  sec  69°  15'.3  =  0.4507,    log  esc  60°  57'.9  =  0.0583. 

(§  8.)  The  first  page  of  this  table  (p.  7)  contains  the  log 
sin  and  log  tan  for  every  minute  from  0°  to  6°,  the  degrees 
and  names  being  given  at  the  top,  and  the  minutes  in  the 
left-hand  column,  marked  '  at  the  top,  and  the  log  cos  and 
log  ctn  for  every  minute  from  84°  to  90°,  the  degrees  and 
names  being  given  at  the  bottom,  and  the  minutes  in  the 
right-hand  column  marked  '  at  the  bottom.  Only  the  third 
and  fourth  decimal-figures  of  the  log  tan  (or  log  ctn)  are 
printed,  the  preceding  figures  being  the  same  as  for  the 
log  sin  (or  log  cos),  except  when  the  third  decimal-figure 
of  the  log  tan  (or  log  ctn)  is  printed  in  small  type,  in 
which  case  the  second  decimal-figure  is  to  be  increased  by 
1.     Thus:  — 

log  sin  1°  23'=  8.3828,  log  cos  86°  37'  =  8.7710, 

log  tan  1°  23'  =  8.3829,  log  ctn  86°  37'  =  8.7717, 

log  sin  4°  33'  =  8.8994,  log  cos  87°  40'  =  8.6097, 

logtan  4°  33'=8.9008,  log  ctn  87°  40'  =  8.6101. 

"When  the  angle  contains  a  fraction  of  a  minute,  inter- 
polation may  be  employed ;  and  the  numbers  printed  in 
small  type  after  the  values  of  the  functions  are  multipliers 
(derivatives),  which  may  be  used  instead  of  the  tabular 
differences  through  half  the  intervals  before  and  after  the 
line  on  which  they  stand,  and  for  both  the  functions  in  that 
line.  For  example,  to  find  the  log  sin  and  log  tan  of 
lp  03'.6  (==  1°  04'  —  0'.4),  we  have 


66  THE   ELEMENTS   OF   LOGAEITHMS. 

0.4  X  68  =  27.2,  log  sin  1°  03'.6  =  8.2699  —  27  =  8.2672, 
log  tan  1°  03.6  =  8.2700  —  27  =  8.2673. 

In  like  manner  :  — 

log  sin  1°  22'.2  =  8.3786,  log  cos  87°  37'.8  =  8.6165, 
log  tan  1°  22'.2=  8.3787,  log  eta  87°  37'.8=  8.6169. 

Where  the  printed  multiplier  is  omitted,  as  in  the 
upper  part  of  the  first  column  of  the  table,  interpolation 
cannot  properly  be  applied  to  the  table,  and  the  method 
which  will  be  explained  in  (§  10)  must  be  used  for  angles 
not  tabulated. 

The  log  esc  and  log  ctn  of  angles  between  0°  and  6^ 
and  the  log  sec  and  log  tan  of  angles  between  84°  and  90° 
can  be  found  by  (A).     Thus  we  have 

log  esc  1°  22'.2  =  1.6214,  log  sec  87°  37'.8  =  1.3835, 
log  ctn  1°  22'.2  =  1.6213,  log  tan  87°  37'.8  =  1.3831. 

(§  9.)  The  right-hand  division  of  page  7  is  a  table  for 
finding  the  log  sec  or  log  cos  of  any  angle  below  9°  51'  or 
the  log  esc  or  log  sin  of  any  angle  above  80°  09'.  The 
first  column  of  this  division,  marked  "  Angles  "  at  the  top, 
contains  those  small  angles,  expressed  to  the  nearest  tenth 
of  a  minute,  at  which  the  last  figure  of  the  log  sec  and  log 
cos  expressed  to  four  decimal  places  changes  its  value. 
The  second  column,  marked  "  Angles  "  at  the  bottom,  con- 
tains the  complementary  angles  of  the  former.  In  the 
intermediate  lines,  and  in  the  third  column,  marked  "sec" 
at  the  top  and  "esc"  at  the  bottom,  are  the  third  and 
fourth  decimal  figures  of  the  log  sec  of  the  angles  which 
lie  between  those  named  in  the  first  column,  and  of  the 
log  esc  of  the  angles  between  those  named  in  the  second 
column;  the  preceding  figures,  which  are  always  0.00, 
being  given  at  the  top  and  bottom  of  the  column.  Thus, 
0.0006  is  the  log  sec  (to  four  places)  of  any  angle  between 
2°  53'.0  and  3°  08'.0  and  the  log  esc  of  any  angle  between 
80°  52'.0  and  879  07\0. 


APPENDIX.  G7 

The  arithmetical  complement  of  the  tabulated  loga- 
rithm is  the  log  cos  or  log  sin  of  the  given  angle.  Thus, 
we  have 

log  sec  1°  22'.2  =  0.0001,  log  esc  87°  37'.8  =  0.0004, 
log  cos  1°  22fJ2  =  9.9999,  log  sin  87°  37/.8  =  9.9996. 

This  little  table  leads  us  to  observe  the  slow  rate  of 
variation  of  the  cos  and  sec  when  their  angle  is  near  0°, 
and  of  the  sin  and  esc  when  their  angle  is  near  90°. 
Hence  a  small  angle  cannot  be  very  accurately  found  from 
its  cos  or  sec,  or  an  angle  near  90°  from  its  sin  or  esc ;  for 
the  function,  or  its  logarithm,  expressed  to  a  given  number 
of  places,  may  correspond  to  any  angle  included  icithin  a 
considerable  range.     See  also  (§  17). 

0  10.)  The  log  sin,  log  tan,  log  esc,  and  log  ctn  of 

small  angles  and  the  log  cos,  log  ctn,  log  sec,  and  log  tan 
of  angles  near  90°  can  also  be  found  by  the  sin  ail  tables 
at  the  bottom  of  the  table  of  Logarithms  of  Numbers, 
which  correspond  to  that  explained  in  (§  4).  We  must  find 
the  logarithm  of  the  given  angle  (or  of  its  complement  in 
the  case  of  an  angle  near  90°)  in  minutes,  and  for  this 
purpose  we  may  generally  use  the  extreme  left-hand 
column  of  the  table  of  logarithms,  headed  "Angles," 
without  reducing  the  angle  to  minutes.     Thus:  — 

log  sin  4°  23'.6  =  2.4209  +  6.4633  =  8.8842, 
log  tan  4°  23'.6  =  2.4209  +  6.4646  =  8.8855, 
log  sin  1°  03'.6  =  1.8035  +  6.4637  =  8.2672, 
log  tan  1°  03'.6  =  1.8035  +  6.4638  =  8.2673, 
log  cos  87°  37'.8  =  log  sin  2°  22'.2 

=  2.1529  +  6.4636  ==  8.6165, 
log  ctn  87°  37'.8  =  2.1529  +  6.4640  =  8.6169, 
log  ctn  89°  59'.90127  =  log  tan  0°  00'.09873 

=  8.9944  +  6.4637  =  5.4581. 


OS  THE    ELEMENTS    OF    LOGARITHMS. 

(§  11.)  The  functions  of  angles  in  all  quadrants  may- 
be found  by  the  method  which  lias  already  been  explained 
in  (§  5).  The  table  at  the  top  of  page  10,  the  arrangement 
of  which  will  be  readily  understood,  may  be  used  in  this 
connexion.     E.g. :  — 

log  sin    97°  18'.4  ==  9.9964, 

log  ctn  413°  47'.9  =  9.8645, 

log  tan  192°  41'.3  ==  9.3525«, 
log  cos  (—108°  26'.2)  =  9.5001>i, 

log  esc  339°  15'.3  =  0.4507m, 

log  cos  (— .  37°  28'.7)  =  9.8996, 

log  sec  590°  27'.0  =  0.1960;?, 

log  sin  178°  56'.4  =  8.2672. 

(§  12.)  In  astronomy,  angles  are  sometimes  expressed 
in  time,  instead  of  arc,  the  circle  being  divided  into  24 
hours. 

The  angles  in  our  table  are  accordingly  expressed  in 
this  system,  as  well  as  in  degrees  and  minutes.  This  is 
the  case  with  all  the  angles  in  the  principal  table  of  p.  7, 
and  with  the  exact  degrees  on  the  other  pages  of  the 
table,  little  tables  for  converting  minutes  into  time  and 
time  into  minutes  being  given  on  each  of  those  pages,  in 
the  corners  of  the  main  table.  Thus,  we  have 
log  sin  0h  10™08*  =  8.6454,  log  ctn  2h  31»l  15*  =:  0.1101, 
log  esc  bh  33™  20s  =  0.0029,  log  sec  47t  09»l53s  =  0.3352. 

(§  13.)  In  navigation  and  surveying, horizontal  angles 
are  often  expressed  in  points  of  the  compass,  of  which 
eight  constitute  the  quadrant,  and  one  is  equal  to  11°  15'. 
At  the  top  of  page  11  will  be  found  a  table  of  logarithmic 
trigonometric  functions  for  every  quarter-point.  The 
points  at  the  top  are  to  be  used  with  the  names  on  the 
left,  and  the  points  at  the  bottom  with  the  names  on 
the  right.     Thus,  we  have 


APPENDIX.  69 

log  sin  li  pt  ==  9.3856,       log  ctn    5J  pt  =  9.7280, 
log  sec  7  pt  =  0.7098,      log  cos  10$  pt  =  9.6734. 

Interpolation  sliould  not  be  applied  to  the  compass- 
table. 

Inverse  Trigonometric  Functions. 

(§14.)  The  angle  which  corresponds  to  a  given  value 
of  any  logarithmic  trigonometric  function  may  be  found 
by  the  inverse  use  of  the  table  explained  above,  in  (§§  6-13). 
But  a  more  convenient  way  of  arriving  at  the  angle  is 
furnished  by  the  table  of  Inverse  Trigonometric  Func- 
tions. In  the  headings  of  this  table,  A  denotes  any  tri- 
gonometric function,  log  A  its  logarithm,  and  sin- x  A, 
cos-  1A,  &c,  the  angle  of  which  A  is  the  sine,  cosine, 
<&c,  as  the  case  may  be.  This  is  in  accordance  with  cus- 
tom and  with  the  principle  of  notation  explained  in  Loga- 
rithms, §  34.  The  symbol  sin-1  may  be  read  inverse 
sine,  or  antisine,  or  the  angle  (or  arc)of  which  the  sine  is. 
It  is  sometimes  denoted  by  "  arcsin."     Thus  we  have 

log  — 1  =  antilog,     sin  "~1  =  arcsin. 

The  inverse  of  "  log  sin  "  may  be  written  in  either  of  the 
following  ways :  — 

(log  sin)  —  !  =  sin  —  *  log-1  =  arcsin  antilog. 

The  argument  of  this  table  is  the  logarithmic  trigono- 
metric function  (the  characteristic  being  increased  by  10, 
when  negative),  and  is  denoted  by  log  A.  In  the  first 
three  divisions  of  the  table,  occupying  page  12  and  the 
first  half  of  page  13,  the  argument  is  given,  at  intervals 
of  0.01,  from  8.5000  (— 10)  to  0.0000  in  the  left-hand 
column  of  the  table,  and  from  0.0000  to  1.5000  in  the 
right-hand  column  of  the  table.  The  names  at  the  top 
are  to  be  taken  with  the  left-hand  column,  and  the  names 


70  TIIE    ELEMENTS    OF   LOGARITHMS. 

at  the  bottom  with  the  right-hand  column.  Thus,  under 
the  name  sin-1  A,  cos  —  1  A,  tan-1  A,  or  ctn-1  A,  and 
in  the  same  line  with  a  value  of  log  A,  found  at  the  left, 
is  the  angle  of  which  that  value  of  log  A  is  the  log  sin, 
log  cos,  log  tan,  or  log  ctn  ;  and  over  the  name  tan-1  A, 
ctn  —  1  A,  sec-1  A,  or  esc-1  A,  and  in  the  same  line  with 
log  A  at  the  right,  is  the  angle  of  which  log  A  is  the  log 
tan,  log  ctn,  log  sec,  or  log  esc.     E.g. :  — 

(log  sin)-1  8.8200=  3°  47'.3, 
(log tan)-1  9.7400  =  28°  47', 
(log  esc)  -1  0.7800  =  9°  33', 
(log  cos)  -i  9.0700  =  83°  15', 
(log  ctn)  - 1  0.1000  =  38°  28', 
(log  sec)  -1 1.4200  =  87°  49/3. 

The  angle  corresponding  to  a  non-tabulated  value  of 
log^l  may  be  found  by  direct  interpolation;  and  the  num- 
ber in  small  type  on  the  right  or  left  of  any  tabulated  angle 
is  a  multiplier  {derivative)  which  may  be  used,  instead  of 
the  tabular  difference,  in  this  interpolation,  through  half 
the  interval  before  and  after  the  angle  against  which  it 
stands,  being  so  pointed  off  as  to  express  the  number  of 
minutes  by  which  the  angle  is  varied  in  that  part  of  the 
table,  by  a  change  of  one  unit  in  the  third  decimal-place 
of  log  A.  The  general  table  of  Proportional  Parts  (p.  2) 
may  be  used  in  finding  the  correction. 

For  instance,  suppose  we  have  to  find  the  angle  of 
which  the  log  sin  is  8.9023(—  10). 

We  have,  if  log  A  =  8.9000,  sin-1  A  =  4°  33'.4 

by  table  of  Proportional  Parts,         0'.63  X  2.3=         1'.4 
whence,  if  log  A  =  8.9023,  sin  - 1  A  =  4°  34'.8 

adding  so  as  to  approach  the  value  for  8.9100. 

Next,  to  find  (log  ctn)-1  9.7984  (—10)  :  — 


APPENDIX.  71 

We  have,  if  log  A  =  9.8000 

(==  9.7984  +  0.0016),  ctn-1  ^1  =  57°  45'; 

by  table  of  Proportional  Parts,  3'.6  X  1-6  =  6' ; 

whence,  if  log  A  =  9.7984,  ctn  - 1  A  =  57°  51' ; 

again  adding,  so  as  to  approach  the  value  for  9.7900. 

Again,  to  find  (log  sec)—1  0.5157:  — 

We  have,  if  log  A  =  0.5200 

(=0.5157  +  0.0043),  sec-*  .4  =  72°  25'; 

by  table  of  Proportional  Parts,  2'.5  X  4.3  =         11'; 

whence,  if  log  A  =  0.5157,  sec-1  .4  =  72°  14'; 

subtracting  so  as  to  approach  the  value  for  0.5100. 

In  like  manner,  let  the  student  find  the  following 
results : — 

(log  tan)-1  0.6931  =  78°  33', 
(log  esc)-1  1.3574=    2°  31'.0, 
(log  sin)-1  9.0590=    6°  35', 
(log  cos)-1  9.6062=66°  11', 
(log  tan)-1  8.9507=    5°  06'.1, 
(log  ctn)-1  0.5555  =  15°  33'. 

TJie  angle  is,  in  every  case,  to  be  found  icith  the  same 
degree  of  accuracy  as  that  of  the  tabulated  angles  between 
which  it  is  interpolated ;  that  is,  to  the  nearest  tenth  of  a 
minute  if  found  from  the  first  division  of  the  table,  and 
otherwise  to  the  nearest  minute.  The  reason  of  the  dif- 
ference is  that  the  angle  varies  less  rapidly  at  the  begin- 
ning of  the  table  than  at  the  end  for  a  given  change  in 
log  A,  and  hence  is  more  accurately  determined  by  a  value 
of  log  A  given  to  four  places.  Towards  the  end  of  the 
table,  it  is  sometimes  not  even  determined  to  the  nearest 
minute,  as  we  shall  presently  explain  more  particularly. 
See  (§  17). 

(_§  15.)  The  student  should  remark  that  the  angles  in 
the  columns  of  sin ~x  A  and  cos ~x  A  are  complements  of 


72  THE   ELEMENTS    OP   LOGARITHMS. 

each  other;  also  in  those  of  tan-1  A  and  ctn- 1  A; 
and  again  in  those  of  sec-1  A  and  esc-1  A.  This  is  in 
accordance  with  (B). 

Again,  the  values  of  the  argument  on  opposite  sides  of 
the  page  are  arithmetical  complements,  and  therefore  cor- 
respond to  reciprocal  values  of  A ;  while  the  angles  bear 
to  these  values  of  A  reciprocal  relations,  as  the  upper  and 
lower  names  of  the  same  column  (sin-1  A  and  esc-1  A ; 
cos-1  A  and  sec-1  A ;  &c.)  indicate.     See  (A). 

(§  1G.)  The  multipliers  for  the  first  and  second  columns 
are  omitted  in  the  lower  half  of  the  third  division  of  the 
table;  because  in  that  part  of  the  table  they  vary  too 
rapidly  to  admit  of  accurate  interpolation  through  so  wide 
an  interval  as  0.01.  Hence,  the  last  three  divisions  of 
the  table  are  given,  containing  the  argument  at  smaller 
intervals,  from  9.7500  to  0.0000,  the  multipliers  being  still 
so  expressed  as  to  measure  the  variation  in  minutes  for  a 
change  of  0.001  in  the  argument.     Thus,  we  find  :  — 

If  log  sin  cp  =  9.8236,  g>=41°  56'  —  1.4  X  7M  =41°  46' ; 
if  log  sin  cp  =  9.8812,  cp  =  49°  39'  —0.8  X  9.'3=49°  32'; 
if  log  sin  9  =  9.9173,  <?  =  55°  42'  +  0.3  X  12'  =55*  46'. 

The  values  of  cos-1  A  are  the  complements  of  those 
of  sin- 1  A;  and  the  arithmetical  complements  of  the 
tabulated  values  of  log  A  are  the  values  of  the  log  esc  for 
the  angles  in  the  column  sin  —  1  A,  and  of  the  log  sec  for 
the  complements  of  those  angles.     Thus,  we  have :  — 

If  log  cob  <p=z  9.9267,  sin-1  A  =  57°38',g>  =  320  22  ; 
if  log  esc  cp  =  0.0265,  log  sin  cp  =  9.9735,  q,  =  70°  11' ; 
if  log  sec  cp  =  0.1089,  log  cos  cp  =  9.8911,  cp  =  38°  54' ; 

if  log  cos  cp  =  9.8183,  cp  =  48°  51' ; 

if  log  sec  cp  =  0.1252,  cp  =41°  27'; 

if  log  esc  cp  =  0.1879,  cp  =  40°  27' ; 

if  log  sec  cp  =  0.1999,  cp  =  50°  52'. 


APPENDIX.  73 

(§  B7.)  When  a  change  of  0.0001  in  any  logarithmic 
trigonometric  function  produces  a  change  of  more  than  1' 
in  the  corresponding  angle,  the  four-place  logarithm  does 
not  determine  the  logarithm  to  minutes,  since  the  uncer- 
tainty of  ±0.00005  in  the  logarithm  will  leave  an  un- 
certainty of  ±0'.5  in  the  angle.  This  is  never  the  case  with 
the  log  tan  or  log  ctn ;  but  an  angle  can  always  be  found 
to  minutes  from  either  of  these  functions  and  by  means 
of  the  first  three  divisions  of  the  table.  But  with  the 
other  functions  this  uncertainty  may  occur ;  and  the  point 
at  which  it  begins  (at  log  A=  9.8940  or  =  0.1060)  is  indi- 
cated by  a  note  in  the  table.  The  angles  are,  however, 
still  tabulated  to  minutes,  and  the  given  values  may  be 
regarded  as  the  probable  values  for  the  assumed  values  of 
the  functions.  Thus,  if  log  sin  cp  =  9.9900,  of  which  the 
true  value  may  lie  anywhere  between  9.98995  and  9.99005, 
(jp  may  have  any  value  from  77°  43'  to  77°  47'.  But 
if  9.9900  expresses  the  value  of  the  logarithm  exactly, 
cp  =  77°  45',  which  is  therefore  adopted  as  the  probable 
value. 

At  the  bottom  of  the  last  column  of  the  table,  interpo- 
lation becomes  inaccurate.  But  the  uncertainty  just 
spoken  of  is  here  so  great  that  the  angle  cannot  be  deter- 
mined with  nicety  from  the  given  function ;  and  the  inac- 
curacy of  interpolation  is,  therefore,  of  little  consequence. 

It  is  to  be  observed  that,  with  four-place  logarithms, 
the  angle  can  be  determined  to  minutes  from  its  log  tan 
or  log  ctn  in  all  parts  of  the  quadrant ;  from  its  log  sin 
or  log  esc  in  the  first  half  of  the  quadrant ;  and  from  its 
log  cos  or  log  sec  in  the  last  half  This  observation  will 
often  direct  the  computer  in  the  choice  of  his  method, 
where  several  are  open  to  him. 

(§  18.)  This  table  is  only  applicable  to  values  of  the 
logarithm  between  8.5000  (—  10)  and  1.5000.  If  the 
logarithm  lies  outside  of  these  limits,  the  angle  may  be 

7 


74  TIIE    ELEMENTS    OF    LOGARITHMS. 

found  by  the  table  of  Logarithms  of  Numbers,  by  the 
formula  printed  between  the  two  divisions  of  page  12, 
which  is  obtained  by  reversing  those  at  the  bottom  of  pp. 
4  and  5»     For  those  formulas  give 

log  (cp  in  minutes)  =  log  sin  cp  —  S=.  log  tan  cp  —  T. 

Hence,  if  £=  T=  6.4637,  we  have 

log  (cp  in  minutes)  =  log  A  +  3.5363  — 10, 

whether  A  denotes  sin  cp  or  tan  cp.  S  has  always  the 
above  value,  when  log  A  <  8.5000,  but  T  may  reach  the 
value  6.4639,  and  thus  give  a  (generally  unimportant) 
correction  of  our  formula. 

If  the  given  logarithm  is  the  log  cos  or  log  ctn  of  the 
required  angle,  that  angle  will  be  the  complement  of  cp 
determined  as  above.  If  log  A  >  1.5000,  we  must  take 
its  arithmetical  complement,  which  will  be  the  logarithm 
of  the  reciprocal  function,  and  proceed  as  above. 

Thus:  — 

If  log  sin  cp  =  7.5192,         log  <r  =  1.0555,  cp  =  11;.3G  ; 

if  log  cos  cf  ==  8.0079,  log  cog>=  1.5442,  co  qp=  35'.01, 

9  =  89°  24'.99; 
if  1.  >g  tan  cf  =  6.8197,         log  cp  =  0.3560,  cp  =  2'.270 ; 

if  log  ctn  c;  =  1.9981,  log  tan  cp  =  8.0019,  cp  —  34'.53  ; 

if  log  sec  cf  — 1.5019,  log  cos  cp  =  8.4981,  cp  =  8$°  1T.7; 

if  log  esc  qp  =  4.1254,  log  sin  cp  =  5.8746,  cp  =  0'.2576. 


(§  10.")  Acute  angles  only  nre  directly  given  by  this 
table.  When  angles  in  any  quadrant  are  admissible,  the 
signs  of  the  given  trigonometric  functions  must  be  noted, 
and  it  must  be  remembered  that  to  every  value  of  any 
function  two  angles  correspond,  in  any  four  consecutive 
right  angles ;  for  example,  between  0°  and  360°,  or  be- 


ArPEXDix.  75 

tween  —  180°  and  -f-  180°.      The  values  of  the  angles 
may  be  found  from  the  table  by  the  aid  of  (C).     Thus :  — 


(log  sin)-1  8.9023/1  =  184°  34'.8 

or  =  355°25'.2; 

(log  ctn)-1  9.7984  =  57°  51 

or  =  237°  51' ; 

(log  sec)-1  0.5157/1  =  107°  46' 

or=  252°  14'; 

(log  tan)-1  0.6931w  =  101°  27' 

or=  281°  27'; 

(log  cos)-1  9.6062  =  66°  11' 

or  =  —  66°  11' ; 

(log  sin)-1  9.0590  =   6Q  35' 

or  =  173°  25'. 

In  many  questions,  the  required  angle  can  only  be 
acute  or  obtuse.  It  is  then  fully  determined  by  the  sign 
of  the  given  function,  unless  that  function  is  a  sine  or  co- 
secant ;  and,  in  that  case,  two  values  of  the  angle  are 
admissible  if  the  function  is  positive,  none  if  the  function 
is  negative.     Thus,  in  such  a  question :  — 

(log  ctn)-1  0.5555/1  =  164°  27'; 

(log  sin)-1  9.8236    =  41°  46'  or  =  138°  14' ; 

(log  cos)-1  9.9267    =32°  22'' ; 

(log  sin)-1  9.8812/1  is  an  inadmissible  angle. 

(§  SO.-)  This  table  can  also  be  used  inversely  to  find  a 
logarithmic  trigonometric  function  of  a  given  angle  ;  and 
if  we  were  restricted  to  one  trigonometric  table,  it  would 
probably  be  found  more  convenient  to  use  this  table  both 
directly  and  inversely  than  that  which  precedes  it. 

Let  the  student  find  from  this  table  :  — 

logtan57°46'  =0.2003,  log  sec  50°  27'  =0.1961, 
log  cos  63°  56'  =  9.6429,  log  sin  4°  23'.6  =  8.8843, 
log  ctn   4°  23'.6  =  1.1144,  log  ctn  87°  37'.8  =  8.6170. 

(§  SI.")  If  the  natural  value  to  four  figures  of  any  tri- 
gonometric function  of  a  given  angle,  or  the  angle  which 
corresponds  to  such  a  value,  is  required,  we  must  use  the 


76  THE    ELEMENTS    OF   LOGARITHMS. 

table  of  Logarithms  of  Numbers,  in  connexion  with  one 
of  the  four-place  trigonometric  tables.     Thus  :  — 

tan      57°  46'  =  1.586,        sin      4°  23'.6  =  0.0766, 
sec-1  26.30    =  87°  49/.3,  ctn-i  0.8217  =  50°  36'. 


Examples  in  Trigonometry  for  the  Four-Place  Tables. 

Seaver's  Formulas  of  Trigonometry  (Boston,  John  Allyn)  will  be  found 
a  convenient  book  of  reference  in  these  examples. 

(§  22.)  Plane  Bight  Triangles. 
Ex.  1.  Given  :  h  =  0.4958,  A  =  54°  44'. 

A  =0.4958  log  9.6953  9.6953 

A  =  54°  44'  log  sin        9.9120       log  cos  9.7615 


log  a          9.6073 

log  b     9.4568 

i?  =  90°—  .4  =  35°  16'   a=  0.4048 

b  =  0.2863 

Ex.  2.  Given: h  =  54.57,  a  =  23.48. 

A  =  54.57 

colog  8.2630 

a  =  23.48 

log  1.3707 

h  +  a  =  78.05        log  1.8924  jg  ™  J }  9.6337 

h  —  a  —  31.09        log  1.4927  A  =  25°  29' 

2)3.3851  B  =  04°  31' 

log  b     1.6926  b  =49.28 
Ex.  3.  Given  :  a  =  0.04437,  b  =  0.02216. 

a  =  0.04437          log           8.6471  8.6471 
b  =0.02216      colog           1.6544 

A  =  63°  28'          log  tan  )  ~~~  log  esc  ) 

B  =  26°  32'           log  ctn  }  °-3015  log  sec  \  °-0483 

h  =  0.04959  log           8.6954 

We   may  also  find   h  by  the  Gaussian  table,  as  in 
Logarithms,  §  66. 


APPENDIX.  77 

Ex.  4.  Given:  h  =  0.3736,  B=  12°  30'. 
To  be  computed :  A  =  77°  30',  a  =  0.3648,  ft  =  0.08086. 

JSfe  5.  Given:  ft  =  14.548,-5=  54°  24'. 
To  be  computed :  A  =  35°  36',  A  =  17.888,  «  =  10.415. 

Ex.  6.  Given  :  ft  =  11.111,  .4  =  11°  11'. 
To  be  computed:  B=  78°  49',  h  =  11.324,  a  =  2.196. 

JSb.  7.  Given:  7*  =  0.06723,  ft  =  0.05489. 
To  be  computed :  a  =  0.03882,  A  =  35°  16,  B  =  54°  44'. 

£fc  8.  Given :  a  =  8.148,  ft  =  10.864. 
To  be  computed:  h  =  13.58,  ^  =  36°  53',  B  =  53°  07'. 

(_§  23.)    /Special  Cases  of  Plane  Bight  Triangles. 

Seaver's  Formulas,  130. 

Ex.  1.  Given:  h  =  4602.8360,  ft  =  4602.2106. 

h—b=       0.6254  log  9.7962 

2h  =  9205.6720         colog  6.0359 


2)5.8321 

M 

log  sin  7.9160 

3.5363 

£-4=0°  28'.33 

log  1.4523 

yl  =  0°  56'.66 

.5=89*  03'.34 

.2.  Given:  a  =3792.8 

1714,  ft  =  3791.8692. 

a  =  3792.8714 

colog  6.4211 

ft  =  3791.8692 

colog  6.4212 

a_ft=       1.0022 

log  0.0009 

a+  ft  =7584.7406 

log  3.8800 

2. 

colog  9.6990 

^  —  i? 

log  tan  6.4222 

3.5363 

A  —  B  =   0°  00'.9088        log  9.9585 
A  +  B  =  90°  OO'.OOOO 

A  =  45°  00'.4544  B=U°  59'.5456. 


78  THE   ELEMENTS    OF    LOGARITHMS. 

Ex.3.   Given:  h  =  210.54219,  a  =  210.48823. 

h  —  a=     0.05396       log  8.7321 
2  h  =  421.08438     colog  7.3756 

2)6.1077 

45°  —  \  A  log  sin  8.0538 

3.5363 


45° 

-M= 

-    0C 

'  38'.92 

log 

1.5901 

\A: 

=  44c 

'  21'.08 

A-- 

=  88° 

42U6 

B-- 

=  l°17/.84 

Ex.  4. 

Given : 

A  = 

107.9823, 

B  = 

:2°08'.64. 

To  be  computed  :  a  =  107.8867,  b  =  4.040,  A  =  87°  51'.36. 

Ex.5.  Given:  a  =  920139,  5  =  925897. 

To  be  computed:  ^  =  44°  49'.27,  J5  =  45°  10'.73. 

Ex.  6.  Given:  h  =  2.096889,  ^  =  86°  25'.38. 

To  be  computed :  a  =  2.092802,  b  =  0.1308,  B=  3°  34'.62. 

(§  24.)  Plane  Oblique  Triangles. 
Ex.  1.  Given:  a=  0.3578, -Z?=32°41',  C=  47°  54'. 

^1  —  180°  —  (J5+  C)  =  99°  25'. 

a  =0.3578       log  9.5537  9.5537 

B  =  32°  41'      log  sin    9.7324  (7  =  47°  54'  log  sin  9.8704 
A  =  99°  25'      W  esc    0.0059  0.0059 


b  =0.1959       log  9.2920   c  =  0.2691    log        9.4300 

Ex.  2.  Given  :  c=  325.06,  A  =  154°  22',  (7=8°  03'. 
To  be  computed  :  B  =  17°  35',  a  =  1004.3,  5  =  701.2. 
Ex.  3*  Given  :  a  =  4236,  J  =  5124,  B  =  124°  50'. 


*  Each  of  the  examples  3-9  should  be  illustrated  by  a  figure. 
7* 


APPENDIX.  79 

a  =  4*236  log  3.6270 

b  =5124         colog  6.2904        log  3.7096  3.7096 

B  =  124°  50'  log  sin  9.9142  log  esc  0.0858  0.0858 

Ax  =  42°  44' log  sin  9.8316 
A2  =137*16' 
<7i=  12°  26'  log  sin  9.3330 

(72  =—  82°  06'  log  sin  9.9959;i 

log  3.1284       log  3.7913^ 
C!  =  1344    c2=— 6184 

Ex.  4-  Given :  a  =  4236,  b  =  5124,  A  =  124°  50'. 

Compute:  Bx  =  83°  04',   <7i=  —  27°  54',  ci=  — 2416. 
J?2  =  96°  56',  <72  =  —  41°  46',  c2  =  —  3438. 

Ex.5.  Given:  a  =  49.00,  c=  67.98,  (7=32°  18/ 

Compute :  A\  =   22°  39',  Bx  =  125°  03',  h  =  104.17 ; 
A2  =  157°  21',  B2  =  —  9°  39',  62  =  —  21.32. 

JS&.  &  Given:  a  =  49.00,  c  =  49.00,  (7=32°  18'. 

Compute :  Ax  =    32°  18',  Bx  =  115°  24',  bx  =  82.84 ; 
^2=147*42^,^2=     0*00',  52=   0. 

-Sis.  7.  Given:  a  =  49.00,  c  =  31.24,  (7=32°  18'. 

Compute  :  Ax  =    56*  57',  Bx  =  90°  45',  h  =  58.46 ; 
^2  =  123*  03',  B2  =  24*  39',  h  =  24.38. 

^c.5.  Given:  a  =49.00,  c=  26.18,  <7<=32*  18'. 

To  be  computed  :A1=A2  =  90*  00',  i?i  =  B2  =57°  42', 
^  =  h  =  41.42. 

jEk.  9.  Given:  a  =49.00,  c=  12.50,  (7=32*  18'. 

We  find  log  sin  .4  =  0.3211.  .-.  sin  A  >  1,  which  is 
impossible,  and  there  is  no  triangle  which  satisfies  the 
conditions. 

Ex.  10.  Given  :  a  =  6.239,  b  =  2.348,  C=  110*  32'. 


80  THE    ELEMENTS    OF   LOGARITHMS. 

a  +  b  =  8.587  colog  9.0662 

a  —  £  =  3.891  log  0.5900 

90°  —  L(7  =  L(^_L-i?)—  34°  44'    log  tan  9.8409 

\  (A  —  B)  =  17°  26'  log  tan  9.4971 
A  =  52°  10' 

B  =  17°  18'  log  esc  0.5267 

£7=110°  32'  log  sin  9.9715 

b  =  2.348  log  0.3707 

c= 7.395  log  0.8689 

The  side  c  can  also  be  found  in  the  following  ways  :  — 

(a  +  b)  log        0.9338     (a  —  b)     log        0.5900 
i(A  +  B)  log  cos  9.9147  log  sin  9.7557 

x  \a  —  B)  log  sec  0.0204  log  esc  0.5235 

c=  7.395   log        0.8689  c  =  7.400  log        0.8692 
i  (^  —  B)  is  so  small  that  the  slight  error  in  its  com- 
puted value  gives  rise  to  a  perceptible  error  in  its  log  esc, 
and  therefore  in  c  as  computed  by  the  last  formula. 

Ex.  11.  Given  :    a  =  0.01511,    c  =  0.03100,    B  = 
169°  45'. 
To  be  computed :  A  =  3°  21'.3,  (7  =  6°  53'.7,  b  =  0.2541. 

Ex.  12.   Given  :   a  =  0.13561,  b  =  0.12348,  c  = 
0.14091. 

s  =  0.20000  colog  0.6990 

5  —  a  —  0.06439  log  8.8088  co  1.1912 
5  —  5  =  0.07652  log  8.8838  co  1.1162 
5—  c  =  0.05909   log  8.7715  co  1.2285 

2)17.1631 

log  r  8.5816  8.5816 

A  =  61°  18'  \A  —  30°  39'  log  tan  9.7728 
B=  53°    00'  \B  =  26°  30'  log  tan  9.6978 

6  —  65°    42'  \C=  32°  51'  log  tan  9.8101 

2^=180°  00' 


APPENDIX.  81 

Ex.13.  Given:  a  =17.856,  5  =  13.349,  c  =  11.111. 
To  be  computed  :  A  =  93°  20',  B  =  48°  16',  C=  38°  24'. 

Ex.  U.  Given:  a  =  32.571,  A  =29°  47'.61,  J?  = 
33°  10'.37. 

Compute  by  Seaver,  131,  Third  Method:  5  =  35.868. 

Ex.  15.  Given :  C=  121*  42',  log  a  =  1.8968,  log  6 
=  2.0014. 
Compute  by  Seaver,  133,  Second  Method :  ^  =  25°  20', 
^=32°  58'. 

Ex.16.  Given:  a  =  0.062387,   b  =  0.023475,    C  = 
110°  32'. 
Compute  by  Seaver,  133,  Third  Method :  c  =  0.07395. 

(§  25.)  Heights  and  Distances. 
Ex.  1.  At  the  distance  of  277.3  feet  from  a  tower,  which 
stands  on  a  horizontal  plane,  the  angle  of  elevation  of  the 
tower  is  observed  to  be  37°  28'.    Find  its  height. 

Ans.  217.5  feet. 

Ex.  2.  From  the  top  of  a  tower  367.6  feet  high,  an 
object  is  observed  in  the  horizontal  plane  on  which  the 
tower  stands,  with  an  angle  of  depression  of  49°  43'. 
Find  the  distance  of  the  object  from  the  foot  of  the 
tower.  Ans.  311.4  feet. 

Ex.  3.  Two  observers,  stationed  on  opposite  sides  of 
a  cloud,  observe  the  angles  of  elevation  to  be  51°  23'  and 
30°  18',  their  distance  apart  being  1000  feet.  Find  the 
height  of  the  cloud  and  its  distances  from  the  two  ob- 
servers. Ans.  2712  feet ;  3472  feet ;  5401  feet. 

Ex.  4.  The  angle  of  elevation  of  the  top  of  a  tower, 
which  stands  on  a  horizontal  plane,  is  observed  at  one 
station  to  be  68°  19',  and  at  another  station,  546  feet  farther 
from  the  tower,  to  be  32°  34'.     Find  the  height  of  the 


82  THE    ELEMENTS    OF    LOGARITHMS. 

tower,  and  its  distances  from  the  two  points  of  observa- 
tion. Ans.  467.6  feet ;  185.9  feet ;  731.9  feet. 

Ex.  5.  An  observer  on  shipboard  sees  a  cape  bearing 
N.N.E. ;  and  after  sailing  30  miles  N.W.  by  N.,  he  sees  the 
same  cape  bearing  E.  by  S.  Find  the  distances  of  the  cape 
from  the  two  points  of  observation. 

Ans.  21.65  miles ;  25.43  miles.  This  example  is  most 
easily  solved  by  finding  the  angles  in  points  (namely,  5 
pts,  4  pts,  and  7  pts),  and  then  using  the  table  at  the  top 
of  p.  11  of  the  Tables. 

Ex.  6.  Coasting  along  shore,  I  saw  a  cape  of  land 
bearing  N.N.E. ;  and  after  sailing  W.N.W.  20  miles,  I 
saw  it  bearing  N.E.  by  E.  Find  the  distances  of  the  cape 
from  the  ship  at  both  stations. 

Ans.  29.93  miles ;  36.00  miles. 

Ex.  7.  Being  at  sea,  we  saw  two  headlands,  of  which 
one  bore  S.W.  by  W.  and  the  other  W.  byN.  The  chart 
showed  that  the  first  headland  bore  S.E.  from  the  second, 
and  was  distant  from  it  23.25  miles.  Find  our  distances 
from  both  headlands.        Ans.  18.26  miles ;  32.25  miles. 

Ex.  8.  Two  ships  sail  from  the  same  port,  the  one  S.W. 
30  miles,  and  the  other  S.E.  by  S.  40  miles.  Find  the  bear- 
ing and  distance  of  the  second  ship  from  the  first. 

Ans.  S.  74°  30'  W. ;  45.08  miles. 

Ex.  9.  An  observer  from  a  ship  saw  two  headlands ; 
the  first  bearing  N.E.  by  E.,  and  the  second  N.W.  After 
he  had  sailed  N.N.W.  10.25  miles,  the  first  headland  bore 
E.  by  N.,  and  the  second  W.N.W.  Find  the  bearing  and 
distance  of  the  first  headland  from  the  second. 

Ans.  N.  88°  02'  E.,  or  nearly  due  east ;  35.25  miles. 

Ex  10.  An  observer  saw  two  headlands;  the  first 
bearing  S.E.,  the  second  E.S.E.  After  sailing  E.  by  N. 
10  miles,  he  saw  the  first  bearing  S.  by  E.,  and  the  second 


ArrENDix.  83 

S.E.  by  S.     Find  the  bearing  and  distance  of  the  first 
headland  from  the  second.  Arts.  S.S.W.  6.89  miles. 

Ex.  11.  At  one  station,  the  bearing  of  a  cloud  is 
N.N.W.,  and  its  angle  of  elevation  50°  35'.  At  a  second 
station,  bearing  from  the  first  N.  by  E.  and  distant  1  mile, 
the  bearing  of  the  cloud  is  W.  by  N.  Find  the  height  of 
the  cloud  and  its  distance  from  each  station. 

Am.  7727  feet;  10002  feet;  8838  feet. 

Ex.  12.  In  the  midst  of  a  level  plain,  which  is  crossed 
by  a  straight  road,  stands  a  tower,  250  feet  high.  An  ob- 
server at  the  top  of  the  tower  sees  an  object  which  moves 
on  the  road.  At  first,  it  bears  N.N.W.,  and  its  angle  of 
depression  is  16°  08' ;  five  minutes  later,  it  bears  E.  by  S., 
and  its  angle  of  depression  is  32°  18'.  Find  the  direction 
of  the  road,  its  distance  from  the  tower,  and  the  rate  at 
which  the  object  is  moving. 

Am.  S.E.  i  S.;  250.9  feet ;  2.575  miles  per  hour. 

(§  20.)  Areas  of  Triangles. 

Ex.  1.  Given :  a  =  12.34  chains,  b=  17.97  chains,  (7= 
135°  04. 

To  be  computed  :  Area  =  7.832  acres  =  7 A  3R  13r. 
Ex.  2.  Given  :  a—  17.95  chains,  _Z?  =  100°,  (7=70°. 
To  be  computed  :  Area  =  85.90  acres  =  85A  3R  24r. 

Ex.  3.  Given:  a  =  45.56  chains,  5  =  52.98  chains,  c  = 
61.22  chains. 

To  be  computed  :  Area  =  117.3  acres  =  117A  1.2R. 

Ex.  4-  Given:  a=  32.56  chains,  5  =  57.84  chains,  c  = 
44.44  chains. 

To  be  computed:  Area=  71.93  acres  =  71 A  3R  29r. 


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